CN100490334C - Method for constructing and selecting irregular LDPC code based on GRS - Google Patents

Abstract

Construction of LDPC codes using GRS code. A novel approach is presented by which a GRS code may be employed to generate a wide variety of types of LDPC codes. Such GRS based LDPC codes may be employed within various types of transceiver devices implemented within communication systems. This approach may be employed to generate GRS based LDPC codes particular designed for various application arenas. As one example, such a GRS based LDPC code may be specifically designed for use in communication systems that operate in accordance with any standards and/or recommended practices of the IEEE P802.11n Task Force.

Description

Make up and select method based on the GRS abnormal LDPC code

Technical field

The present invention relates to communication system, more specifically, the present invention relates to the encoding and decoding of communication system internal information.

Background technology

Data communication system sustainable development for many years, in recent years, adopting the communication system of repetitive error correcting encoder is the focus that researchers pay close attention to.Wherein of greatest concern is the communication system that adopts low-density checksum (LowDensity Parity Check is called for short LDPC) sign indicating number.Under same state of signal-to-noise, the error rate of the communication system of repeated encoding is usually less than the selective coding.

One of this field continues and main developing direction is that signal to noise ratio essential in reducing communication system is to reach the specific error rate.Desirable target is the mountain farming limit (Shannon ' slimit) of attempting in the research communication channel, and mountain farming limit can be regarded as the data transmission rate that is used in the channel with specific signal to noise ratio, can realize not having the error code transmission by this channel.In other words, farming limit in mountain is the theoretical limit of the channel capacity of given modulation and encoding rate.

The extraordinary decoding performance that can provide in some cases near mountain farming limit has been provided the LDPC sign indicating number.In theory, the LDPC sign indicating number is proved the performance that can reach from 0.004 decibel of mountain farming limit.Length is that 1,000,000 abnormal LDPC code once reached this performance, and it has confirmed that it is very likely using the LDPC sign indicating number in communication system.

Some alternative type of codings that constantly occur in the prior art and modulation scheme also can provide approaching capacity to realize error correction.The LDPC sign indicating number provides such performance, and it is the most probable selection of this technology sustainable development.

The best way that does not have now to follow is set up well behaved LDPC sign indicating number, in the reference below [a], has set up regular LDPC sign indicating number based on two coded words of Reed one Saloman (Reed-Solomon is called for short RS) sign indicating number.

[a]I.Djurdjevic，J.Xu，K.Abdel-Ghaffar and S.Lin，“A Class ofLow-Density Parity-Check Codes Constructed Based on Reed-Solomon CodesWith Two Information Symbols，”IEEE Communications Letter，vol.7，no.7，pp.317-319，July 2003.

Yet the LDPC code type that uses this prior art to propose is very narrow, and makes the flexibility of the LDPC sign indicating number that designs other types in this way very little.Lacking flexibility all is a significant challenge for the communication equipment that designs this LDPC sign indicating number and/or use this LDPC sign indicating number to realize.Clearly, under different state of signal-to-noise, be used for various communication systems and continue with the coding techniques demand of better type that the better error correction average and the error rate are provided.

Summary of the invention

The present invention relates to arithmetic facility and method, and partly provide further instruction at ensuing description of drawings, embodiment and claims.

According to an aspect of the present invention, provide the method for a kind of structure based on the abnormal LDPC code of general Reed one Saloman (Generalized Reed-Solomon is called for short GRS), described method comprises:

For the LDPC code block is selected a plurality of possible bit degree distribution;

Select to obtain a bit degree distribution of the top performance limit in described a plurality of possible bit degree distribution based on the performance of determining by the density evolution method;

Based on selected bit degree distribution, will be decomposed into a plurality of submatrixs corresponding to parity matrix based on GRS rule LDPC sign indicating number, each submatrix in wherein said a plurality of submatrixs has corresponding bit degree, and has corresponding a plurality of permutation matrixes; And

Replace at least one permutation matrix at least one submatrix in described a plurality of submatrix so that at least one submatrix includes null matrix and permutation matrix with null matrix, thereby generate corresponding based on the GRS abnormal LDPC code.

Preferably, described method further comprises:

Each bit degree distribution in described a plurality of possible bit degree distribution is carried out the analysis of density evolution method, thereby generate the density evolution method analysis result of a plurality of correspondences; And

By more described density evolution method analysis result, from described a plurality of possible bit degree distribution, select a bit degree distribution.

Preferably, described a plurality of submatrix is three submatrixs.

Preferably, described method further comprises:

Replace at least one permutation matrix in first submatrix in described three submatrixs with null matrix;

Replace in described three submatrixs at least one permutation matrix in second submatrix with null matrix.

Preferably, described method further comprises:

Generation and at least the second is based on corresponding at least the second parity matrix of GRS abnormal LDPC code;

To being applied to described in the communication channel, determine in first performance aspect the Block Error Rate of the function of signal to noise ratio based on the GRS abnormal LDPC code;

To being applied to described at least the second in the described communication channel, determine in second performance aspect the Block Error Rate of the function of signal to noise ratio based on the GRS abnormal LDPC code;

When described first performance has the wrong rank lower than described second performance (error floor), select described based on the GRS abnormal LDPC code;

When described second performance has than the lower wrong rank of described first performance, select described at least the second based on the GRS abnormal LDPC code.

Preferably, the described bit degree distribution of selecting from described a plurality of possible bit degree distribution comprises three bit degree.

Preferably, each permutation matrix in described a plurality of permutation matrix is 81 * 81 matrix.

Preferably, described null matrix is that 81 * 81 element is 0 matrix.

Preferably, described have minimum intersymbol distance based on GRS rule LDPC sign indicating number; Described also have minimum intersymbol distance based on the GRS abnormal LDPC code.

Preferably, the size in each loop of corresponding described LDPC two-dimensional plot (bipartitegraph) based on the GRS abnormal LDPC code is at least 6.

Preferably, described method further comprises:

Structure is corresponding to the generator matrix of described parity matrix, wherein said parity matrix corresponding to described based on the GRS abnormal LDPC code;

Use described generator matrix that at least one information bit is encoded, thereby generate at least one LDPC code word of ldpc coded signal.

Preferably, described method further comprises:

Use is corresponding to described described parity matrix based on the GRS abnormal LDPC code, to using described ldpc coded signal to decode,, at least one information bit of encoding in the described ldpc coded signal estimates thereby being made the best based on GRS abnormal LDPC code coding.

According to an aspect of the present invention, provide the method for a kind of selection based on the GRS abnormal LDPC code, described method comprises:

Make up with a plurality of based on the corresponding a plurality of parity matrixs of GRS abnormal LDPC code;

Determine that in a plurality of performances aspect the Block Error Rate of the function of signal to noise ratio described a plurality of performances are corresponding with described a plurality of each based on the GRS abnormal LDPC code that are used for communication channel;

From described a plurality of based on selecting to have of the lowest error rank determined according to described a plurality of performances the GRS abnormal LDPC code based on the GRS abnormal LDPC code, described selection have corresponding parity matrix based on the GRS abnormal LDPC code.

Preferably, described method further comprises:

Consideration will be used for the complexity according to described a plurality of first first decoders of decoding based on GRS abnormal LDPC code encoded signals based on the GRS abnormal LDPC code;

Consideration will be used for the complexity according to described a plurality of second second decoders of decoding based on GRS abnormal LDPC code encoded signals based on the GRS abnormal LDPC code;

When the complexity of described first decoder is lower than described second decoder, select described first based on the GRS abnormal LDPC code;

When the complexity of described second decoder is lower than described first decoder, select described second based on the GRS abnormal LDPC code.

Preferably, the parity matrix in described a plurality of parity matrix comprises first submatrix and second submatrix; At least one permutation matrix in described second submatrix is replaced by null matrix.

Preferably,

First parity matrix in described a plurality of parity matrix by with form based on the corresponding parity matrix of GRS rule LDPC sign indicating number;

Second parity matrix in described a plurality of parity matrix also by with form based on the corresponding described parity matrix of GRS rule LDPC sign indicating number;

Corresponding described parity matrix based on GRS rule LDPC sign indicating number comprises first submatrix and second submatrix, and described first submatrix and described second submatrix include corresponding a plurality of permutation matrixes;

At least one permutation matrix in described first submatrix is substituted by null matrix, thereby generates described first parity matrix in described a plurality of parity matrix;

At least one permutation matrix in described second submatrix is substituted by null matrix, thereby generates described second parity matrix in described a plurality of parity matrix.

Preferably,

Each parity matrix in described a plurality of parity matrix by with form based on the corresponding parity matrix of GRS rule LDPC sign indicating number;

Described have minimum intersymbol distance based on GRS rule LDPC sign indicating number;

Described a plurality of each based in the GRS abnormal LDPC code also has minimum intersymbol distance.

Preferably, a plurality ofly be at least 6 with described based on one in the GRS abnormal LDPC code each loop size based on the corresponding LDPC two-dimensional plot of GRS abnormal LDPC code.

Preferably, described method further comprises:

Make up with described corresponding to the corresponding generator matrix of selecting of the parity matrix based on the GRS abnormal LDPC code;

Use described generator matrix that at least one information bit is encoded, thereby generate at least one LDPC code word of ldpc coded signal.

Preferably, described method further comprises:

Use with selected based on the corresponding described parity matrix of GRS abnormal LDPC code, selectedly decode to using, thereby at least one information bit of encoding in the described ldpc coded signal is made best estimation based on GRS abnormal LDPC code encoded signals.

According to an aspect of the present invention, provide a kind of structure corresponding to the method based on the parity matrix of GRS abnormal LDPC code, described method comprises:

Based on bit degree distribution, will be decomposed into a plurality of submatrixs based on the corresponding parity matrix of GRS rule LDPC sign indicating number, each submatrix in wherein said a plurality of submatrix has corresponding bit degree, and each submatrix in described a plurality of submatrix has corresponding a plurality of permutation matrixes;

Substitute at least one permutation matrix at least one submatrix in described a plurality of submatrixs with null matrix, thus generate with based on the corresponding parity matrix of GRS abnormal LDPC code.

Preferably, described method further comprises:

For LDPC sign indicating number module is selected a plurality of possible bit degree distribution;

Select described bit degree distribution from described a plurality of possible bit degree distribution, wherein selected bit degree distribution has the best performance limit in described a plurality of possible bit degree distribution; And wherein

Described have minimum intersymbol distance based on GRS rule LDPC sign indicating number;

Described also have minimum intersymbol distance based on the GRS abnormal LDPC code.

Preferably, be at least 6 with described size based on each loop in the corresponding LDPC two-dimensional plot of GRS abnormal LDPC code.

Preferably, described method further comprises:

Generate with at least one additional based on corresponding at least one the additional parity matrix of GRS abnormal LDPC code;

According to the Block Error Rate that is the function of signal to noise ratio,, determine first performance to being applied to described in the communication channel based on the GRS abnormal LDPC code;

According to the Block Error Rate that is the function of signal to noise ratio,, determine second performance to being applied to described at least one the additional abnormal LDPC code in the described communication channel based on GRS;

When described first performance has than the lower wrong rank of described second performance, select described based on the GRS abnormal LDPC code;

When described second performance has than the lower wrong rank of described first performance, select that described at least one is additional based on the GRS abnormal LDPC code.

Preferably, described method further comprises:

Make up with corresponding to the described corresponding generator matrix of described parity matrix based on the GRS abnormal LDPC code;

Use described generator matrix that at least one information bit is encoded, thereby generate at least one LDPC code word of ldpc coded signal.

Preferably, described method further comprises:

Use with described based on the corresponding described parity matrix of GRS abnormal LDPC code, to using described ldpc coded signal to decode,, at least one information bit of encoding in the described ldpc coded signal estimates thereby being made the best based on GRS abnormal LDPC code coding.

According to below in conjunction with the accompanying drawing detailed description of the present invention, other features and advantages of the present invention will be apparent.

Description of drawings

The invention will be further described below in conjunction with drawings and Examples, in the accompanying drawing:

Fig. 1 is the schematic diagram of an embodiment of communication system;

Fig. 2 is the schematic diagram of an embodiment of communication system;

Fig. 3 is the schematic diagram of an embodiment of LDPC sign indicating number two-dimensional plot;

Fig. 4 is the schematic diagram of structure based on an embodiment of the method for GRS abnormal LDPC code;

Fig. 5 is the schematic diagram of selection based on an embodiment of the method for GRS abnormal LDPC code;

Fig. 6 is to use corresponding to the parity matrix based on GRS rule LDPC sign indicating number and generates and schematic diagram based on an embodiment of the corresponding parity matrix of GRS abnormal LDPC code;

Fig. 7 is on additional white Gaussian noise (AWGN) communication channel, based on GRS rule LDPC sign indicating number (LDPC ₀) and second based on GRS abnormal LDPC code (LDPC ₂) between carry out the schematic diagram of a performance embodiment relatively;

Fig. 8 is on the AWGN communication channel, based on GRS abnormal LDPC code (LDPC ₂) and a selectable LDPC sign indicating number (LDPC _(a)) between carry out the schematic diagram of a performance embodiment relatively;

Fig. 9 is the schematic diagram of bit to an embodiment of the interleaving technique of symbol;

Figure 10 is the schematic diagram of bit to an embodiment of the interleaving technique of symbol;

Figure 11 is the schematic diagram of bit to an embodiment of the interleaving technique of symbol;

Figure 12 is the schematic diagram of bit to an embodiment of the interleaving technique of symbol;

Figure 13 is the schematic diagram of bit to an embodiment of the interleaving technique of symbol;

Figure 14 is in Rayleigh decay communication channel, based on GRS rule LDPC sign indicating number (LDPC ₀),

First based on GRS abnormal LDPC code (LDPC ₁) and second based on GRS abnormal LDPC code (LDPC ₂) between carry out the schematic diagram of a performance embodiment relatively;

Figure 15 is in channel, first based on GRS abnormal LDPC code (LDPC ₁), second based on GRS abnormal LDPC code (LDPC ₂) and an optional LDPC sign indicating number (LDPC _(b)) between carry out the schematic diagram of a performance embodiment relatively;

Figure 16 is the schematic diagram of bit to an optional embodiment of the interleaving technique of symbol;

Figure 17 is the schematic diagram of bit to an optional embodiment of the interleaving technique of symbol;

Figure 18 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944, the 937) sign indicating number (1) and the first coding LDPC (c) (1944,972);

Figure 19 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944, the 1297) sign indicating number (2) and the second coding LDPC (d) (1944,1296);

Figure 20 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944,487) sign indicating number (3) and the 3rd coding LDPC (e) (1944,487);

Figure 21 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944,1621) sign indicating number (4) and the 4th coding LDPC (f) (1944,1620);

Figure 22 A is the schematic diagram of an embodiment that generates the method for ldpc coded signal;

Figure 22 B is the schematic diagram of an embodiment of method that ldpc coded signal is decoded;

Figure 23 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944, the 1296) sign indicating number (5) and the second coding LDPC (d) (1944,1296);

Figure 24 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944,486) sign indicating number (6) and the 3rd coding LDPC (e) (1944,486);

Figure 25 is in communication channel, carries out the schematic diagram of an embodiment of performance comparison between based on GRS abnormal LDPC (1944,1620) sign indicating number (7) and the 4th coding LDPC (f) (1944,1620);

Figure 26 is in communication channel, use dissimilar positions to the interleaving technique of symbol first based on GRS abnormal LDPC code (LDPC ₁), second based on GRS abnormal LDPC code (LDPC ₂) and an optional LDPC sign indicating number LDPC _(b)Between carry out the schematic diagram of a performance embodiment relatively.

Embodiment

The present invention proposes a kind of new method, uses RS sign indicating number or GRS sign indicating number to generate a plurality of abnormal LDPC codes.Make in this way and to generate a large amount of this type of abnormal LDPC codes and provide very big scope for the designer.Some inventor has invented the method for using GRS sign indicating number create-rule LDPC sign indicating number.Using RS sign indicating number or GRS sign indicating number to make up regular LDPC sign indicating number provides best minimum intersymbol apart from estimated value.The wrong rank of this rule-like LDPC sign indicating number occur when lower error rate.Yet well-known, when being used for reaching the channel capacity (or mountain farming limit) of communication channel, regular LDPC sign indicating number does not have abnormal LDPC code good.

In order to make up one on wrong rank with reach aspect the channel capacity two all good LDPC sign indicating numbers of performance, the present invention proposes a kind of new method, makes up irregular LDPC sign indicating number based on RS sign indicating number or GRS sign indicating number.Further part in the application, among the possible embodiment of the present invention, compare with some known abnormal LDPC codes in IEEE (institute of electrical and electronic engineers) 802.11n seminar (that is: the seminar of exploitation 802.11TGn (high-throughput) standard) the recommendation convention of exploitation and the application of standard, this abnormal LDPC code of the present invention produces 0.8 to 1.0dB gain.

To setting up before this LDPC sign indicating number is described in detail, introduce some communication systems and/or the communication equipment that adopts this LDPC sign indicating number earlier, simultaneously the LDPC sign indicating number is provided some brief descriptions.

The target of digital communication system is from a position or but subsystem transmits the numerical data of inerrancy or acceptance error rate to another position or subsystem.As shown in Figure 1, data can be transmitted in various communication systems by various communication channels: magnetic media, wireless network, optical fiber, copper cash and other media.

Illustrated in figures 1 and 2 is to be respectively the schematic diagram of the embodiment of communication system 100 and 200.

Referring to Fig. 1, in the communication system 100, the communication equipment 110 (comprising transmitter 112 with encoder 114 and the receiver 116 with decoder 118) that communication channel 199 will be positioned at communication channel 199 1 ends is connected with another communication equipment 120 that is positioned at communication channel 199 other ends (comprising transmitter 126 with encoder 128 and the receiver 122 with decoder 124).In certain embodiments, communication equipment 110 and 120 all includes only a transmitter or a receiver.Communication channel 199 can realize by several dissimilar media (for example: use satellite antenna 132 and 134 satellite communication channel 130, use communications tower 142 and 144 and/or local antenna 152 and 154 the fiber optic communication channel 160 of radio communication channel 140, electric wire communication system 150 and/or use electric light (E/O) interface 162 and photoelectricity (O/E) interface 164).In addition, can use to combine and form communication channel 199 more than a kind of media.

In order to reduce the undesired error of transmission that occurs in the communication system, adopt error correction and channel coding schemes through regular meeting.In general, these error corrections and channel coding schemes use encoder at transmitter terminal, use decoder at receiver end.

Referring to communication system shown in Figure 2 200, transmitting terminal in communication channel 299, information bit 201 is provided for transmitter 297, transmitter 297 uses encoder with symbol mapper 220 (this symbol mapper can be considered different separately functional blocks 222 and 224) information bit 201 to be encoded, thereby generate centrifugal pump modulation symbol sequence 203 and offer emission driver 230, this emission driver 230 uses digital to analog converter 232 to generate and transmits 204 continuously, and transmitting 205 continuously after using emission filter 234 to generate filtering, it is 205 abundant consistent with communication channel 299 to transmit continuously after this filtering.At the receiving terminal of communication channel 299, received signal 206 is provided for AFE (analog front end) 260 continuously, and this AFE (analog front end) 206 comprises receiving filter 262 (continuous received signal 207 after the generation filtering) and analog to digital converter 264 (generating discrete received signal 208).Measure maker 270 and generate symbol tolerance 209, decoder 208 uses this symbol to measure 209 pairs of described centrifugal pump modulation symbols and makes best estimation (210) with the information bit that is encoded in wherein.

Communication equipment in the previous embodiment can use various decoding feature described herein to realize.In addition, ensuing several accompanying drawing is described (some embodiment will provide more detailed introduction) to other and the specific embodiment that is used to support to realize equipment, system, function to the decoding of ldpc coded signal and/or method.Before proposing introduction more specifically, earlier the LDPC sign indicating number is carried out general introduction.

Fig. 3 is the schematic diagram of an embodiment of LDPC sign indicating number two-dimensional plot 300.In the prior art, the LDPC two-dimensional plot is also referred to as Tan Natu (TannerGraph) sometimes.The LDPC sign indicating number can be considered as having the code of binary parity check matrix, and like this, nearly all element value is 0 (for example: this binary parity check matrix is a sparse matrix) in this matrix.For example, H=(h _{I, j}) _{N * M}Can be counted as block length is the parity matrix of the LDPC sign indicating number of N.

In this parity matrix, 1 quantity can be expressed as d in the i row _v(i), 1 the quantity that j is capable can be expressed as d _c(j).If for all i, d _v(i)=dv, and for all j, d _c(j)=d _c, this LDPC sign indicating number is called as (d so _v, d _c) regular LDPC sign indicating number, otherwise this LDPC sign indicating number is called abnormal LDPC code.

Provided introduction in the reference [2] that reference [1] that LDPC sign indicating number R.Gallager is below shown and M.Luby etc. are shown.

[1] R.Gallager, Low-Density Parity-Check Codes, Cambridge, MA:MITPress, 1963. (R.Gallager, low density parity check code, Cambridge, MA:MIT publishing house, 1963)

[2] M.Luby, M.Mitzenmacher, M.A.Shokrollahi, D.A.Spielman, and V.Stemann, " Practical Loss-Resilient Codes ", Proc.29th Symp.onTheory ofComputing, 1997, pp.150-159. (M.Luby, M.Mitzenmacher, M.A.Shokrollahi, D.A.Spielman, and V.Stemann, " practical elastic extenuation code ", the 29th theory of computation seminar collection of thesis in 1997,150-159 page or leaf)

Rule LDPC sign indicating number can be expressed as two-dimensional plot 300, the left node of its parity matrix is represented bits of coded (or " variable node " (or " bit node ") 310 decoded to ldpc coded signal with the method for position decoding) variable, and the right side node is represented verification equation (or check-node 320).Two-dimensional plot 300 by the LDPC sign indicating number of H definition can be defined by N variable node (for example N bit node) and M check-node.Each variable node in N variable node 310 has d _v(i) individual sideline (edge, label 330 is depicted as one of them example) is with bit node v _i312 link to each other with one or more check-nodes (M check-node in).Sideline 330 shown in the figure is with bit node v _i312 with check-node c _j322 connect.Sideline d _vQuantity (as d among the figure _vShown in 314) be called the degree (degree) of variable node i again.Approx, each check-node of M check-node 320 has just d _c(j) individual sideline (is expressed as d among the figure _c324) this node is connected with one or more variable nodes (or bit node) 310.The quantity d in this sideline _cThe degree that is called check-node j again.

At variable node v _i(or bit node b _i) 312 and check-node c _jSideline 330 between 322 may be defined as e=(i, j).But on the other hand, (i, j), the node in this sideline also can be represented as e=(v (e), c (e)) (perhaps e=(b (e), c (e)) for given sideline e=.Given variable node v _i(or bit node b _i), can define one group from node v _i(or bit node b _i) sideline of sending is that Ev (i)={ e|v (e)=i} (perhaps is E _b(i)={ e|b (e)=i}).Given check-node c _j, can define one group is E from the sideline that node cj sends _c(j)={ e|c (e)=j}.Therefore, derivation result is | E _v(i)=d _v| (or | E _b(i) |=d _b) and | E _c(j) |=d _c

As a rule, anyly can be characterized as graphical nodes by the node that two-dimensional plot is represented.It is also to be noted that abnormal LDPC code also can be described with two-dimensional plot.Yet the degree of each group node can be selected according to some distribution in abnormal LDPC code.Therefore, for two of an abnormal LDPC code different node v _I1And v _I2, | E _v(i ₁) | may be not equal to | E _v(i ₂) |.This relation is also set up for two check-nodes.The notion of abnormal LDPC code provides introduction by people such as M.Luby at first in reference [2].

In a word, by the two-dimensional plot of LDPC sign indicating number, the parameter of LDPC sign indicating number can be defined by the degree that distributes, and this point has provided description at people's such as above-mentioned M.Luby reference [2] and below with reference to data [3]:

[3]T.J.Richardson and R.L Urbanke，“The capacity of low-density paritycheck code under message-passing decoding，”’IEEE Trans.Inform.Theory，Vol.47，pp.599-618，Feb.2001.

This distribution can be described to:

If λ _iFor from spending the mark in the sideline of sending, establish ρ for the variable node of i _iBe the mark in the sideline of sending for the check-node of i from degree, so, spend the distribution group (λ, ρ) may be defined as follows:

$\mathrm{\λ}\left(x\right)=\underset{i=2}{\overset{M_y}{\mathrm{\Σ}}}{\mathrm{\λ}}_1{x}^{i-1}$ With $\mathrm{\ρ}\left(x\right)=\underset{i=2}{\overset{M_c}{\mathrm{\Σ}}}{\mathrm{\ρ}}_i{x}^{i-1},$ M wherein _vAnd M _cThe maximal degree of representing variable node and check-node respectively.

At the beginning, introduce some information, help reader understanding's RS sign indicating number used according to the invention and GRS sign indicating number to make up abnormal LDPC code about RS sign indicating number and GRS sign indicating number.

Finite field

Suppose that finite field (Galois Field, Galois Field) is GF (p ^m), wherein p is a prime number.If α is the primitive element in this territory, so,

$\mathrm{GF}\left({\mathrm{\ρ}}^m\right)=\{0,\mathrm{\α}...,{\mathrm{\α}}^{p^m-1}\}$ (equation 1)

2 code words by two-dimentional RS sign indicating number generation

If ρ≤p ^m-1.If C is a length is the Simplified two-dimension RS sign indicating number of ρ.So as can be known, the minimum intersymbol distance of this RS sign indicating number is ρ-2+1=ρ-1.In addition, this coding right of possession heavy (being the quantity of non-0 element) is the code word of ρ or ρ-1.A kind of possible method that makes up this coding provides in below with reference to data [a] (quoting in the introduction in front), and this method provides introduction below.

[a]I.Djurdjevic，J.Xu，K.Abdel-Ghaffar and S.Lin，“A Class ofLow-Density Parity-Check Codes Constructed Based on Reed-Solomon CodesWith Two Information Symbols，”IEEE Communications Letter，vol.7，no.7，pp.317-319，July 2003.

Define a multinomial g (x) ∈ GF (p ^m) [x], so

$g\left(x\right)=(x-\mathrm{\α})(x-{\mathrm{\α}}^2)...(x-{\mathrm{\α}}^{\mathrm{\ρ}-2})=\underset{i=0}{\overset{\mathrm{\ρ}-2}{\mathrm{\Σ}}}{g}_i{x}^i$ (equation 2)

G wherein _ρ-2=1.So, utilize this multinomial, can generate two-dimensional encoded with following generator matrix.

$G=\left(\begin{array}{cccc}{g}_0& {\mathrm{<figure-callout\; id="1"\; label="g"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">g</figure-callout>}}_1...g_{\mathrm{\&rho;}-3}& 1& 0\\ 0& g_0& g_{\mathrm{\&rho;}-3}& 1\end{array}\right)$ (equation 3)

Next, (for example: c from this two-dimensional encoded C, take out weight ρ code word ₀) and weight ρ-1 code word (for example: c ₁).

Selectively, two code words in the GRS sign indicating number can generate according to the content of following U.S. Provisional Patent Application and U.S. Patent Application Publication, and following patent application is quoted by present patent application at this.

1, (case number is acted on behalf of to denomination of invention by office: BP4008.2) for the U.S. Provisional Patent Application of " using the RGS sign indicating number to make up the LDPC sign indicating number ";

2, (case number is acted on behalf of to denomination of invention by office: BP4372) for the U.S. Patent application of " using the RGS sign indicating number to make up the LDPC sign indicating number ".

Make simple introduction in the selectable method that these two code words to this use GRS sign indicating number generate the LDPC sign indicating number.

In the GRS sign indicating number, integer ρ can be 1 to p ^mBetween any numeral.Get group of addresses $L=\{{\mathrm{\&alpha;}}^{i_0},...,{\mathrm{\&alpha;}}^{i_{\mathrm{\&rho;}-1}}\}.$ From Galois Field (is GF (p ^m)) in get ρ non-0 element v ₀, v ₁... v _ρ-1Can generate a following two-dimentional GRS sign indicating number so:

$C=\left\{(v_{0}f\left({\mathrm{\&alpha;}}^{i_0}\right),v_{1}f\left({\mathrm{\&alpha;}}^{i_1}\right),...,v_{\mathrm{\&rho;}-1}f\left({\mathrm{\&alpha;}}^{i_{\mathrm{\&rho;}-1}}\right))\right|f\&Element;\mathrm{GF}\left(p^m\right)\left[x\right],\mathrm{deg}\left(f\right)<2\}$ (equation 4)

GF (p wherein ^m) [x] be that Galois Field (is GF (p ^m)) on polynomial ring.Degree of getting is 1 polynomial f ₀=f _0,1X+f _0,0And f ₁=f _1,1X+f _1,0, f wherein _{I, j}∈ GF (p ^m), like this, for all λ ∈ L, f ₀(λ) ≠ 0, and for all β ∈ GF (p ^m), f ₁(x) ≠ β f ₀(x).Two code words of among the C this can be expressed as so:

$c_0=(v_0{f}_0\left({\mathrm{\&alpha;}}^{i_0}\right),v_1{f}_0\left({\mathrm{\&alpha;}}^{i_1}\right),...,v_{\mathrm{\&rho;}-1}{f}_0\left({\mathrm{\&alpha;}}^{i_{\mathrm{\&rho;}-1}}\right))$

$c_1=(v_0{f}_1\left({\mathrm{\&alpha;}}^{i_0}\right),v_1{f}_1\left({\mathrm{\&alpha;}}^{i_1}\right),...,v_{\mathrm{\&rho;}-1}{f}_1\left({\mathrm{\&alpha;}}^{i_{\mathrm{\&rho;}-1}}\right))$ (equation 5)

Two code words by two-dimentional RS sign indicating number generation

Two code words by coding C (are c ₀, c ₁), can generate an one dimension RS sign indicating number and p ^m-1 coset (coset).

First one-dimensional coding can following method produce:

C ₀={ β c ₀| β ∈ GF (p ^m)={ c _0,0, c _0,1..., c _{0, pm-1}(equation 6)

Another p ^m-1 coset can following method produce:

C _i=α ^I-1c ₁+ C ₀={ α ^I-1c ₁+ x|x ∈ C ₀, i=1 ..., p ^m-1 (equation 7)

Each coset C _iCan be expressed as $C_i=\{{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">\; <figure-callout\; id="0"\; label="c"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">c</figure-callout>\; </figure-callout>}}_{\mathrm{1,0}},...,c_{i,p}{\mathrm{}}^m-1\}.$ In addition, each ρ-vectorial c _{I, j}Can be expressed as c _{I, j}=(c _{I, j, 0}..., c _{I, j, ρ-1}), c wherein _{I, j, k}∈ GF (p ^m).

The regular LDPC sign indicating number that generates by the word of coset

Definition position figure L:GF (p ^m) → { 0,1}p ^m, L (α like this ⁱ) be a p ^m-vector, and this i+1 is 1, and all other positions are 0.For example, L (0)=(10...0), L (α)=(010...0) or the like.

For each coset C _i, can set up ρ independent ρ ^m* ρ ^m-permutation matrix is as follows:

$\mathrm{Pi},k=\left(\begin{array}{c}L\left(c_{i,0,k}\right)\\ L\left(c_{i,1,k}\right)\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ L\left(c_{i,p^m-2,k}\right)\\ L\left(c_{i,p^m-1,k}\right)\end{array}\right),k=0,...,\mathrm{\&rho;}-1$ (equation 8)

Select one group of γ coset, such as Parity check matrix H can be established as follows:

(equation 9)

This matrix is a low density matrix.Therefore, can use this low density matrix to generate LDPC sign indicating number (with the LDPC matrix of this low density matrix) as the LDPC sign indicating number.Clearly, to have bit degree be that γ, verification degree are the two-dimensional plot of ρ for such LDPC sign indicating number.And such LDPC sign indicating number is regular LDPC sign indicating number.

The reference of being shown according to the Djurdjevic that mentions before etc. [a] as can be known, when γ was even number, the minimum intersymbol of this LDPC sign indicating number was apart from being at least γ+2; When γ is odd number, be γ+1.In other words, the minimum intersymbol of this LDPC sign indicating number is apart from d _MinCan be expressed as:

This type of LDPC sign indicating number can be described as based on the LDPC sign indicating number of RS or based on the regular LDPC sign indicating number of GRS.Because the RS sign indicating number is a kind of of GRS sign indicating number, the further part of present specification generally uses term GRS sign indicating number to represent, is considered as comprising various RS sign indicating numbers and GRS sign indicating number.

Structure is based on the abnormal LDPC code of GRS

In order to realize approaching channel capacity (mountain farming limit) and lower wrong rank simultaneously, the present invention proposes a kind of new method at this, by adjusting regular LDPC sign indicating number, make up abnormal LDPC code based on GRS based on GRS according to the principle structure of H in the above-mentioned equation 9.Corresponding to this based on the generation of the parity matrix of the abnormal LDPC code of GRS can by replace with full null value matrix with the corresponding parity matrix of the regular matrix based on GRS that makes up according to above-mentioned equation 9 in the method for some permutation matrix realize; This replaces the method for permutation matrix to can be described as " puncture (puncturing) " with the null value matrix.That is to say that at least one permutation matrix in this parity matrix is replaced by null matrix (promptly all elements by 0 matrix).

A kind of selection of design is to replace this permutation matrix with null matrix (promptly all elements are 0 matrix), leaves the very big scope of designer for, selects to replace which permutation matrix by null matrix.

Fig. 4 is the schematic diagram of structure based on an embodiment of the method 400 of GRS abnormal LDPC code.

Shown in step 410 among the figure, described method is selected a plurality of possible bit degree distribution for the LDPC code block.According to prior art as can be known, (for example, in abnormal LDPC code) uses 3 different bit degree can obtain best performance in the LDPC code block.Yet, it should be noted that the present invention can also adopt the bit degree distribution that comprises more than 3 kinds different bit degree, and do not depart from the scope of the present invention and spirit.3 different bit degree distribution have been adopted among several embodiment described herein.

Then, in the step 420, from described a plurality of possible bit degree distribution, select a bit degree distribution.It should be noted that selected bit degree distribution has the best performance limit in described a plurality of possible bit degree distribution.In some cases, can from described a plurality of possible bit degree distribution, select a bit degree distribution based on the performance of determining by the density evolution method, shown in step 422.

For example, select to adopt which bit degree distribution can use density evolution method theories and methods to realize, to obtain this bit degree distribution.Provided detailed elaboration in the density evolution method reference [3] (quote the front) below.

[3]T.J.Richardson and R.L Urbanke，“The capacity of low-density paritycheck code under message-passing decoding，”’IEEE Trans.Inform.Theory，Vol.47，pp.599-618，Feb.2001.

A given encoding rate is at first selected γ.Then, parity check matrix H can be configured to the γ p according to the form of equation 9 descriptions ^m* ρ p ^mMatrix.The maximal bit degree of abnormal LDPC code based on GRS yet to be built is γ.The designer also can select other bit degree less than γ, and the bit degree distribution of selecting their correspondences based on other principle.Quantity with bit in the code block of same bits degree will be the multiple of pm.

Below provide an example selecting a plurality of possible bit degree distribution.

Example 1: establish p=3, m=4, ρ=24 and γ=8.Regular LDPC sign indicating number based on GRS can be constructed as 648 * 1944 H matrix so, comprises 192 81 * 81 different permutation matrixes.Its bit degree is 8, and the verification degree is 24.According to what mention before, use 3 different bit degree abnormal LDPC code that can offer the best usually.In ensuing example, selecting the lowest bit degree is 2.Usually, the lowest bit degree in the bit degree distribution can be any less than 8 number.In all possible bit degree distribution of LDPC code block, the bit degree distribution of considering in this particular example comprises 3 different bit degree distribution.Consider 11 possible bit degree distribution in particular for the LDPC code block.This 11 possible bit degree distribution have been shown in the following table:

	Degree=8	Degree=7	Degree=6	Degree=5	Degree=4	Degree=3	Degree=2
								D1	648					648	648
D2	648				648		648
								D3	324				972		648
D4	162				1134		648
								D5	486				810		648
D6	648			648			648
								D7	216			1080			648
D8	432			864			648
								D9	648		648				648
D10	324		972				648
								D11	648	648					648

Table 1

The density evolution method of introducing in the above-mentioned references of mentioning that the people showed such as use Richardson can find that bit degree distribution D3 is one of best candidate's bit degree distribution.Usually, can use any method from a plurality of possible bit degree distribution, to select a bit degree distribution.For example, can use the bit degree distribution choice criteria that in a plurality of possible bit degree distribution, has the best performance limit.Perhaps, can adopt the density evolution method to select to be used to make up bit degree distribution based on the abnormal LDPC code of GRS.

The parity matrix of corresponding bit degree distribution D3 can be expressed as H.This parity check matrix H comprises 8 * 24 independent 81 * 81 different submatrixs (being called transposed matrix when not replaced by null matrix).Parity check matrix H has the permutation matrix of 8 elements of 4 row, the permutation matrix of 4 elements of 12 row and the permutation matrix of 2 elements of 8 row.Remaining submatrix is null matrix (being the matrix that all elements is null value) entirely.Therefore, having only 4 * 8+12 * 4+8 * 2=96 submatrix is permutation matrix.192-96=96 permutation matrix need be replaced by null matrix in initial regular LDPC sign indicating number.

As mentioned above, described method is selected a bit degree distribution from described a plurality of possible bit degree distribution in step 420.The selection which or which permutation matrix is replaced by null matrix has very big design limit of consideration.For example, after having made up a plurality of different abnormal LDPC codes based on GRS, performance that need to consider various abnormal LDPC codes based on GRS (for example, selection can provide of top performance), the complexity that the decoder of under certain application cases this code signal being decoded is realized, and other does not depart from the scope of the present invention the Consideration with spirit.It should be noted that the designer need carry out a large amount of emulation when determining which LDPC sign indicating number based on GRS can offer the best performance.

Example 1 (continuing)

Below continue to introduce above-mentioned example 1, to show the several possible selection of correspondence based on the parity matrix of GRS abnormal LDPC code.Subsequently, provide several performances relatively, so that the improvement on the performance that employing obtains based on the GRS abnormal LDPC code to be shown.

Described then method based on the bit degree distribution of having selected, is divided into a plurality of submatrixs (each submatrix has corresponding bit degree) with correspondence based on the parity matrix of GRS rule LDPC sign indicating number in step 430.The quantity of the bit degree in the quantity of described submatrix and the bit degree distribution of having selected is corresponding.

In a possible embodiment of the present invention, when selected bit degree distribution comprises 3 independent bit degree, this parity check matrix H will be broken down into 3 independently submatrixs.Continue this example, this decomposed parity check matrix H can be expressed as:

H=[H ₁, H ₂, H ₃] (equation 10)

When considering the above-mentioned bit degree distribution D3 that has selected, each submatrix all has the bit degree of a correspondence.For example, according to above-mentioned table 1, submatrix H ₁Bit degree be 8; Submatrix H ₂Bit degree be 4; Submatrix H ₃Bit degree be 2.The first submatrix H ₁A kind of possible design can be expressed as:

$\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}1=\left(\begin{array}{cccc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,1}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,2}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,3}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,4}}\\ {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,1}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,2}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,3}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,4}}\\ {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,1}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,2}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,3}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,4}}\\ {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,1}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,2}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,3}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,4}}\\ {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,1}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,2}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,3}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,4}}\\ {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,1}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,2}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,3}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,4}}\\ P_{\mathrm{7,1}}& P_{\mathrm{7,2}}& P_{\mathrm{7,3}}& P_{\mathrm{7,4}}\\ P_{\mathrm{8,1}}& P_{\mathrm{8,2}}& P_{\mathrm{8,3}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{8,4}}\end{array}\right)$ Displacement

This first submatrix H ₁Be by single permutation matrix P _Ij648 * 423 matrixes that constitute, each independent transposed matrix P _IjIt is 81 * 81 permutation matrix.Though it should be noted that and adopt multiple different value in the various examples of this application, clearly, a lot of particular values described here can be selected by the designer based on the GRS abnormal LDPC code and revise, and design the coding that is fit to application-specific with this.In other words, these values only are used to help reader understanding's various aspects of the present invention, and the designer is free to adopt the different abnormal LDPC code based on GRS of other value design.

In the step 440, described method is replaced at least one permutation matrix at least one submatrix in described a plurality of submatrixs with null matrix (being that all elements is 0 matrix), thereby generates based on the GRS abnormal LDPC code.

The method of replacing certain or a plurality of permutation matrixes with null matrix has a lot.

The second submatrix H ₂(be expressed as after the modification

) a kind of possible method for designing as follows:

${\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1=\left(\begin{array}{cccccccccccc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,5}}& & & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,8}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,9}}& & & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,12}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,13}}& & & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,16}}\\ {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,5}}& & & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,8}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,9}}& & & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,12}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,13}}& & & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,16}}\\ {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,5}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,6}}& & & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,9}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,10}}& & & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,13}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,14}}& & \\ {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,5}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,6}}& & & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,9}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,10}}& & & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,13}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,14}}& & \\ & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,6}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,7}}& & & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,10}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,11}}& & & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,14}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,15}}& \\ & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,6}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,7}}& & & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,10}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,11}}& & & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,14}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,15}}& \\ & & P_{\mathrm{7,7}}& P_{\mathrm{7,8}}& & & P_{\mathrm{7,11}}& P_{\mathrm{7,12}}& & & P_{\mathrm{7,15}}& P_{\mathrm{7,16}}\\ & & P_{\mathrm{8,7}}& P_{\mathrm{8,8}}& & & P_{\mathrm{8,11}}& P_{\mathrm{8,12}}& & & P_{\mathrm{8,15}}& P_{\mathrm{8,16}}\end{array}\right)$

Second submatrix that this revised

Be 648 * 972 matrix, one 81 * 81 null matrix (for example, wherein all 81 * 81 elements are 0) is represented in each room in the matrix, remaining matrix P _{I, j}All corresponding is permutation matrix.

The second submatrix H ₂(be expressed as after the modification

) another optionally design is as follows:

${\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2=\left(\begin{array}{cccccccccccc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,5}}& & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,7}}& & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,6}}& & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,9}}& & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,11}}& & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,13}}& \\ & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,6}}& & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,5}}& & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,8}}& & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,10}}& & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,12}}& & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,14}}\\ {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,5}}& & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,7}}& & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,6}}& & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,9}}& & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,11}}& & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,13}}& \\ & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,6}}& & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,5}}& & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,8}}& & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,10}}& & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,12}}& & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,14}}\\ {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,5}}& & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,7}}& & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,6}}& & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,9}}& & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,11}}& & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,13}}& \\ & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,6}}& & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,5}}& & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,8}}& & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,10}}& & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,12}}& & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,14}}\\ P_{\mathrm{7,5}}& & P_{\mathrm{7,7}}& & P_{\mathrm{7,6}}& & P_{\mathrm{7,9}}& & P_{\mathrm{7,11}}& & P_{\mathrm{7,13}}& \\ & P_{\mathrm{8,6}}& & P_{\mathrm{8,5}}& & P_{\mathrm{8,8}}& & P_{\mathrm{8,10}}& & P_{\mathrm{8,12}}& & P_{\mathrm{8,14}}\end{array}\right)$

Second submatrix of having revised

This optional embodiment be 648 * 972 matrix, one 81 * 81 null matrix (for example, wherein all 81 * 81 elements are 0) is represented in each room in the matrix, remaining matrix P _{I, j}All corresponding is permutation matrix.

The 3rd submatrix H ₃(be expressed as after the modification

) another kind of possible design as follows:

${\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1=\left(\begin{array}{cccccccc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,17}}& {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,18}}& & & & & & \\ & {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,18}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,19}}& & & & & \\ & & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,19}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,20}}& & & & \\ & & & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,20}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,21}}& & & \\ & & & & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,21}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,22}}& & \\ & & & & & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,22}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,23}}& \\ & & & & & & P_{\mathrm{7,23}}& P_{\mathrm{7,24}}\\ P_{\mathrm{8,17}}& & & & & & & P_{\mathrm{8,24}}\end{array}\right)$

The 3rd submatrix that this has been revised

First embodiment be 648 * 648 matrix, each empty position is represented one 81 * 81 null matrix (for example, wherein all 81 * 81 elements are 0) in the matrix, remaining matrix P _{I, j}All corresponding is permutation matrix.

The 3rd submatrix H ₃(be expressed as after the modification

) another kind of possible design as follows:

${\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^2=\left(\begin{array}{cccccccc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,17}}& & & & & & & {\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{1,24}}\\ {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,17}}& {\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{2,18}}& & & & & & \\ & {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,18}}& {\mathrm{<figure-callout\; id="3"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{3,19}}& & & & & \\ & & {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,19}}& {\mathrm{<figure-callout\; id="4"\; label="P"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{4,20}}& & & & \\ & & & {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,20}}& {\mathrm{<figure-callout\; id="5"\; label="P"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{5,21}}& & & \\ & & & & {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,21}}& {\mathrm{<figure-callout\; id="6"\; label="P"\; filenames="C200610005277E01771.png,C200610005277E01801.png"\; state="\{\{state\}\}">P</figure-callout>}}_{\mathrm{6,22}}& & \\ & & & & & P_{\mathrm{7,22}}& P_{\mathrm{7,23}}& \\ & & & & & & P_{\mathrm{8,23}}& P_{\mathrm{8,24}}\end{array}\right)$

The 3rd submatrix that this has been revised

Optional embodiment also be 648 * 648 matrix, each empty position is represented one 81 * 81 null matrix (for example, wherein all 81 * 81 elements are 0) in the matrix, remaining matrix P _{I, j}All corresponding is permutation matrix.

Thereby revising the method that these submatrixs generate corresponding parity matrix based on the GRS abnormal LDPC code has a lot.The designer has very big scope to select which permutation matrix to be replaced by null matrix.The various embodiment of the submatrix that these have been revised only illustrate some possible example that can generate parity matrix.

Only use the submatrix of having revised on a small quantity, can generate a large amount of different parity matrixs, as follows:

$H\left(1\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$

$H\left(2\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^2]$

$H\left(3\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$

$H\left(4\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$

Wherein two possible parity matrixs will be described in detail below.

Now, based on the regular LDPC sign indicating number (LDPC of GRS ₀) can set up according to the constraint and the design of above-mentioned equation 9, and can adopt p=3, m=4, ρ=24 and γ=8 of setting in the example 1.This is based on the regular LDPC sign indicating number (LDPC of GRS ₀) encoding rate be 0.67.

Then, it is corresponding to GRS abnormal LDPC code (LDPC to use amended submatrix as follows to make up ₁) parity matrix:

$H\left(1\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 11)

This is based on GRS abnormal LDPC code (LDPC ₁) encoding rate also be 0.667.

Then, use amended submatrix as follows can make up corresponding another based on GRS abnormal LDPC code (LDPC ₂) parity matrix:

$H\left(3\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 12)

Should be based on GRS abnormal LDPC code (LDPC ₂) encoding rate also be 0.667.

How the following drawings is depicted as in design process from a plurality of based on selecting the schematic diagram based on the embodiment of GRS abnormal LDPC code the GRS abnormal LDPC code.

Figure 5 shows that and select a schematic diagram based on an embodiment of the method 500 of GRS abnormal LDPC code.Described method generates corresponding a plurality of a plurality of parity matrixs based on GRS abnormal LDPC code (for example, a plurality of distortion of parity matrix) in step 510.Above-mentioned various embodiment can be used to generate corresponding a plurality of described a plurality of parity matrixs based on the GRS abnormal LDPC code.Then, shown in step 520 among the figure, described method determines that described a plurality of each based in the GRS abnormal LDPC code has the performance based on the GRS abnormal LDPC code of corresponding parity matrix (for example, according to the error rate and/or as the Block Error Rate of the function of signal to noise ratio).Then, in step 530, described method from described a plurality of based on select the GRS abnormal LDPC code optimum performance have corresponding parity matrix based on the GRS abnormal LDPC code.In some cases, this optimum performance can have based on the standard on the lowest error rank of the error rate and/or Block Error Rate based on the GRS abnormal LDPC code according to which and selects.Which should select also need the consideration that adds based on the GRS abnormal LDPC code or optionally consider the degree of the difficulty that decoder under the certain application cases realizes/easily, shown in step 532 among the figure.

Figure 6 shows that and use corresponding parity matrix to generate the schematic diagram of correspondence based on the embodiment 600 of the parity matrix of GRS abnormal LDPC code based on GRS rule LDPC sign indicating number.This figure can help the reader understanding to make up the mode of parity matrix.

Visual by figure, corresponding parity check matrix H based on GRS rule LDPC sign indicating number _Reg610, be broken down into a plurality of submatrixs and (be submatrix H shown in the figure ₁621, submatrix H ₂622, submatrix H ₃623, the rest may be inferred, submatrix H _n629).Corresponding parity check matrix H based on GRS rule LDPC sign indicating number _RegIn the quantity of the submatrix that is decomposed can select by designer based on the GRS abnormal LDPC code.

Then, at least one submatrix (for example, submatrix H ₃623) be modified by the mode of replacing at least one permutation matrix wherein with null matrix (value that is all elements is zero matrix); This submatrix of having revised is called submatrix (H3) ' 633.Equally, any one submatrix can be modified; Submatrix H shown in this figure ₃623 are modified to the just possible design alternative of submatrix (H3) ' 633.Obviously, the modification that other submatrixs also can electing property.In addition, more than one submatrix can do not depart from the scope of the present invention with the spirit situation under make amendment.

Use these submatrixs generate corresponding parity check matrix H based on GRS abnormal LDPC code thereafter, _Irr, shown in label 640.

Should be based on the minimum intersymbol of GRS abnormal LDPC code apart from d _MinBe expressed as follows:

It should be noted that shown in label 699, should be based on the minimum intersymbol of GRS abnormal LDPC code apart from d _MinBe used to generate this based on the GRS abnormal LDPC code based on the minimum intersymbol of GRS rule LDPC sign indicating number apart from d _MinIdentical.In other words, based on the GRS abnormal LDPC code with have identical minimum intersymbol apart from d based on GRS rule LDPC sign indicating number _MinEqually, shown in label 699, each is not less than 6 based on the GRS abnormal LDPC code with based on the loop size of GRS rule LDPC sign indicating number.All not having size in based on GRS abnormal LDPC code and the LDPC two-dimensional plot based on GRS rule LDPC sign indicating number in correspondence is 4 loop.

Equally, should be based on the circulation (or loop) that is not less than or equal to 4 in the pairing two-dimensional plot of GRS abnormal LDPC code.The minimal circulation (or loop) of the LDPC two-dimensional plot that this is corresponding is 6.That is to say that the size in each loop of corresponding LDPC two-dimensional plot based on the GRS abnormal LDPC code is at least 6, it is 4 loop that this coding does not have size.

In addition, if should be actually " irregular " LDPC sign indicating number based on the GRS abnormal LDPC code, it will provide than regular LDPC sign indicating number more performance.

In this application, by Block Error Rate (BLER) contrast E _b/ N ₀(energy per bit compares E _bWith spectral noise density N ₀Ratio) various performance map have been described.Block Error Rate is in being usually used in radio communication, and when any one bit was considered to make mistakes in the code block, so whole code block was confirmed as makeing mistakes.In other the communication system applications, the error rate (BER) contrasts E at some _b/ N ₀Can represent its performance.For digital communication system, this term E _b/ N ₀It is the measured value of signal to noise ratio.Check these performance curves, Block Error Rate can be by any given E _b/ N ₀(or signal to noise ratio) determine, thereby the performance-relevant Succinct representation with described coding/decoding method is provided.

Several different performance comparison below are provided, illustrate with other coding and compare, by the performance improvement that provides based on the GRS abnormal LDPC code.

Figure 7 shows that based on GRS rule LDPC sign indicating number (LDPC ₀) (shown in the number in the figure 710) with second based on GRS abnormal LDPC code (LDPC ₂) (shown in the number in the figure 720) performance on AWGN communication channel schematic diagram of an embodiment of 700 relatively.

Binary phase shift keying modulation method and AWGN communication channel have been considered in this first example.These performance curves shown in the figure show, in Block Error Rate=1.5 * 10 ^-5The time, LDPC ₂Performance surpass LDPC ₀1.2dB.

Figure 8 shows that based on GRS abnormal LDPC code (LDPC ₂) (shown in the number in the figure 820) and the performance of LDPC sign indicating number LDPC (a) (shown in the number in the figure 810) on the AWGN communication channel schematic diagram of an embodiment of 800 relatively arbitrarily.

At LDPC sign indicating number and the LDPC shown in Fig. 7 and Fig. 8 ₂All have corresponding as the parity matrix in the above-mentioned equation 12.In addition, should be based on GRS abnormal LDPC code (LDPC ₂) encoding rate also be 0.667.

LDPC will encode among Fig. 8 ₂With encoding rate be that 2/3 coding LDPC (a) compares.This encoding rate is in 2/3 coding (LDPC (a)) reference [4] below introduction to be arranged.

[4]LDPC code motion for Mon 28 Feb 2005 Telecon，WWiSE consortium.

WwiSE (World Wide Spectrum Efficiency, global frequency spectrum rate) is each company of development IEEE802.11n WLAN standard and the alliance that entity is formed.More public informations of relevant WwiSE can obtain from following website:

“http://www.wwise.org/”

Ensuing performance curve shows, when Block Error Rate=1.5 * 10 ^-5Situation under, LDPC ₂Performance exceeds optional LDPC sign indicating number LDPC (a) 0.55dB.

In IEEE 802.11n uses, adopt Rayleigh (Rayleigh) decay communication channel and 64 quadrature amplitude modulations modulation, and the reflection method that provides in the IEEE 802.11a standard, described in reference [5].

[5]Part11：Wireless LAN Medium Access Control(MAC)and PhysicalLayer(PHY)specifications：High-speed Physical Layer in the 5 GHZBand，IEEE Std 802.11a-1999.

The full content of above-mentioned reference [5] is disclosed to the public, and can download from the following network address:

“http://standards.ieee.org/getieee802/download/802.11a-1999.pdf”

Fig. 9, Figure 10, Figure 11, Figure 12 and Figure 13 shows that the schematic diagram of bit to each embodiment of the interleaving technique of symbol.Specifically, Figure 9 shows that embodiment 900 (interlocking 2, is (п 2) shown in the figure); Figure 10 shows that embodiment 1000 (interlocking 3, is (п 3) shown in the figure); Figure 11 shows that embodiment 1100 (interlocking 4, is (п 4) shown in the figure); Figure 12 shows that embodiment 1200 (interlocking 5, is (п 5) shown in the figure); And Figure 13 shows that embodiment 1300 (staggered 6, be (п 6) shown in the figure).Each embodiment is depicted as the intersymbol error of 6 bits that carry out on the LDPC code block of coded-bit (for example LDPC code word).Obviously, the present invention can also adopt the row of any other quantity (being n) to carry out 1 bit interlocking to the n bit.

Referring to the embodiment among Fig. 9 900 (staggered 2, be (п 2) shown in the figure), the LDPC code block 909 that receives can be counted as to be cut apart or is divided into a plurality of parts.For example, LDPC code block 909 is divided into the 0th part 910, part 1 911, part 2 the 912, the 3rd part the 913, the 4th part 914 and the 5th part 915, and each part is provided for corresponding row.

As shown in the figure, each described part is provided for a plurality of row.6 labels (shown in number in the figure 919) as mapping symbols are extracted out from each row, its left side is highest significant position (Most Significant Bit is called for short MSB), and its right side is least significant bit (Least Significant Bit, be called for short LSB), as follows:

First 6 label: c ₀c _kc _2kc _3kc _4kc _5k

Second 6 label: c ₁c _K+1c _2k+1c _3k+1c _4k+1c _5k+1

N 6 label: c _K-1c _2k-1c _3k-1c _4k-1c _5k-1c _6k-1

Each part that offers each respective column has the k position, and the total bit of LDPC code block 909 is 6k.

By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 929) that described each several part is divided into, there is not column permutation.Respectively by the position (c shown in label 932 and 931 _5k-1..., c _4k+1, c _4k) and least significant bit (c _6k-1..., c _5k+1, c _5k) be the redundant digit of from LDPC code block 909, selecting.

Referring to the embodiment among Figure 10 1000 (staggered 3, be (п 3) shown in the figure), the LDPC code block 1009 of reception can be counted as to be cut apart or is divided into a lot of parts.For example, LDPC code block 1009 is divided into the 0th part 1010, part 1 1011, part 2 the 1012, the 3rd part the 1013, the 4th part 1014 and the 5th part 1015, and each part is provided for corresponding row.

The same with aforesaid embodiment, each part is provided for the row of a plurality of same format.Yet as shown in the figure, the order of row is changed.

Six labels (shown in label 1019) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c _4kc _2kc ₀c _5kc _3kc _k

Second 6 label: c _4k+1c _2k+1c ₁c _5k+1c _3k+1c _K+1

N 6 label: c _5k-1c _3k-1c _K-1c _6k-1c _4k-1c _2k-1

Each part that offers each respective column has the k position, and the total bit of LDPC code block 1009 is 6k.

By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1029) that described each several part is divided into, there is column permutation.Respectively by the MSB position (c shown in label 1032 and 1031 _5k-1..., c _4k+1, c _4k) and position (c _6k-1..., c _5k+1, c _5k) be the redundant digit of from LDPC code block 1009, selecting.

Referring to the embodiment among Figure 11 1100 (staggered 4, be (п 4) shown in the figure), the LDPC code block 1109 of reception can be counted as to be cut apart or is divided into a lot of parts.For example, LDPC code block 1109 is divided into the 0th part 1110, part 1 1111, part 2 the 1112, the 3rd part the 1113, the 4th part 1114 and the 5th part 1115, and each part is provided for corresponding row.

The same with aforesaid embodiment, each part is provided for the row of a plurality of same format.Yet as shown in the figure, the order of row is changed.

Six labels (shown in label 1119) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c _3kc ₀c _4kc _2kc _kc _5k

Second 6 label: c _3k+1c ₁c _4k+1c _2k+1c _K+1c _5k+1

N 6 label: c _4k-1c _K-1c _5k-1c _3k-1c _2k-1c _6k-1

Each part that offers each respective column has the k position, and the total bit of LDPC code block 1109 is 6k.

By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1129) that described each several part is divided into, there is column permutation.Respectively by the position (c shown in label 1132 and 1131 _5k-1..., c _4k+1, c _4k) and LSB position (c _6k-1..., c _5k+1, c _5k) be the redundant digit of from LDPC code block 1109, selecting.

Referring to the embodiment shown in Figure 12 1200 (staggered 5, be (п 5) shown in the figure), the LDPC code block 1209 of reception can be counted as to be cut apart or is divided into a lot of parts.For example, LDPC code block 1209 is divided into the 0th part 1210, part 1 1211, part 2 the 1212, the 3rd part the 1213, the 4th part 1214 and the 5th part 1215, and each part is provided for corresponding row.

The same with aforesaid embodiment, each part is provided for the row of a plurality of same format.Yet as shown in the figure, the order of row is changed.

Six labels (shown in label 1219) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c _2kc ₀c _4kc _kc _3kc _5k

Second 6 label: c _2k+1c ₁c _4k+1c _K+1c _3k+1c _5k+1

N 6 label: c _3k-1c _K-1c _5k-1c _2k-1c _4k-1c _6k-1

Each part that offers each respective column has the k position, and the total bit of LDPC code block 1209 is 6k.

By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1229) that described each several part is divided into, there is column permutation.Respectively by the position (c shown in label 1232 and 1231 _5k-1..., c _4k+1, c _4k) and LSB position (c _6k-1..., c _5k+1, c _5k) be the redundant digit of from LDPC code block 1209, selecting.

Referring to the embodiment shown in Figure 13 1300 (staggered 6, be (п 6) shown in the figure), the LDPC code block 1309 that receives can be counted as to be cut apart or is divided into a lot of parts.For example, LDPC code block 1309 is divided into the 0th part 1310, part 1 1311, part 2 the 1312, the 3rd part the 1313, the 4th part 1314 and the 5th part 1315, and each part is provided for corresponding row.

The same with aforesaid embodiment, each part is provided for the row of a plurality of same format.Yet as shown in the figure, the order of row is changed.

Six labels (shown in label 1319) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c ₀c _2kc _4kc _kc _3kc _5k

Second 6 label: c ₁c _2k+1c _4k+1c _K+1c _3k+1c _5k+1

N 6 label: c _K-1c _3k-1c _5k-1c _2k-1c _4k-1c _6k-1

Each part that offers each respective column has the k position, and the total bit of LDPC code block 1309 is 6k.

By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1329) that described each several part is divided into, there is column permutation.Respectively by the position (c shown in label 1332 and 1331 _5k-1..., c _4k+1, c _4k) and LSB position (c _6k-1..., c _5k+1, c _5k) be the redundant digit of from LDPC code block 1309, selecting.

Figure 14 shows that based on GRS rule LDPC sign indicating number (LDPC ₀) (label is 1405), first based on GRS abnormal LDPC code (LDPC ₁) (label is 1410) and second based on GRS abnormal LDPC code (LDPC ₂) (label is 1420) performance on Rayleigh decay communication channel schematic diagram of an embodiment of 1400 relatively.By this embodiment as can be seen, abnormal LDPC ₁And LDPC ₂Under same Block Error Rate, performance exceeds LDPC ₀At least 3dB.

Based on GRS rule LDPC sign indicating number (LDPC ₀) according to the constraint and the design construction of aforesaid equation 9, and can adopt with example 1 in identical value p=3, m=4, ρ=24 and γ=8.Should be based on GRS rule LDPC sign indicating number (LDPC ₀) encoding rate be 0.67.

Corresponding to GRS abnormal LDPC code (LDPC ₁) parity matrix can use the amended submatrix that provides according to above-mentioned equation 11 to make up (, proposing this equation once more) for helping reader:

$H\left(1\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 11)

This is based on GRS abnormal LDPC code (LDPC ₁) encoding rate be 0.667.

Corresponding to GRS abnormal LDPC code (LDPC ₂) parity matrix the modification that provides according to above-mentioned equation 12 can be provided after submatrix make up (, providing this equation once more) for helping reader;

$H\left(3\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 12)

This is based on GRS abnormal LDPC code (LDPC ₂) encoding rate also be 0.667.

Figure 15 shows that first based on GRS abnormal LDPC code (LDPC ₁) (label is 1510), second based on GRS abnormal LDPC code (LDPC ₂) (label is 1520) and the performance of LDPC sign indicating number LDPC (b) (label is 1505) on the communication channel schematic diagram of an embodiment of 1500 relatively arbitrarily.

This embodiment LDPC that will encode ₁1510 and LDPC ₂1520 with encoding rate be that 2/3 coding LDPC (b) 1505 compares.Coding LDPC (b) 1505 has the code length identical with 1944, also is irregular coding.Utilize bit that Fig. 9 provides to the intersymbol error technology, corresponding performance as shown in figure 15.The constructed improvement in performance that 1dB is provided based on the irregular coding of GRS among the figure.

Figure 16 and Figure 17 shows that the schematic diagram of bit to the staggered optional embodiment of symbol.Specifically, Figure 16 and Figure 17 show embodiment 1600 (interlocking 0, is (п 0) shown in the figure) and embodiment 1700 (interlocking 1, is (п 1) shown in the figure) respectively.The same with aforesaid embodiment, each embodiment is depicted as the intersymbol error of 6 bits that carry out on the LDPC code block of coded-bit (for example LDPC code word).Obviously, the present invention can also adopt the row of any other quantity (being n) to carry out 1 bit interlocking to the n bit.

Among the embodiment 1600, LDPC code block 1609 be provided for row in each.First of LDPC code block 1609 offers first row, second of LDPC code block 1609 offers secondary series, the 3rd of LDPC code block 1609 offers the 3rd row, the rest may be inferred, rather than in a plurality of parts of LDPC code block 1609 each is put into corresponding row (as described above described in the embodiment).This shows that shown in label 1629, the order of row is not changed.According to the size (for example, the symbol size of n position) of the symbol that is adopted, as shown in the figure, n+1 symbol offers first row with the program of unrolling.

Six labels (shown in label 1619) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c ₀c ₁c ₂c ₃c ₄c ₅

Second 6 label: c ₆c ₇c ₈c ₉c ₁₀c ₁₁

N 6 label: c _6k-6c _6k-5c _6k-4c _6k-3c _6k-2c _6k-1

The total bit of LDPC code block 1609 is 6k.By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1629) that described each several part is divided into, there is not column permutation.

Among the embodiment 1700, LDPC code block 1709 is provided directly to the every row in a plurality of row.First of LDPC code block 1709 offers first row, second of LDPC code block 1709 offers secondary series, the 3rd of LDPC code block 1709 offers the 3rd row, the rest may be inferred, rather than in a plurality of parts of LDPC code block 1709 each is put into corresponding row (as described above described in the embodiment).This shows that shown in label 1729, the order of row is changed (024135).According to the size (for example, the symbol size of n position) of the symbol that is adopted, as shown in the figure, n+1 symbol offers first row with the program of unrolling.

Six labels (shown in label 1719) as mapping symbols are extracted (its left side is MSB, and its right side is LSB) out from each row as follows:

First 6 label: c ₀c ₂c ₄c ₁c ₃c ₅

Second 6 label: c ₆c ₈c ₁₀c ₇c ₉c ₁₁

N 6 label: c _6k-6c _6k-4c _6k-2c _6k-5c _6k-3c _6k-1

The total bit of LDPC code block 1709 is 6k.By the LSB that from each row, extracts out and MSB as can be seen, with regard to each row (shown in label 1729) that described each several part is divided into, in fact there is column permutation.

In embodiment shown in Figure 16 1600 and embodiment 1700 shown in Figure 17, DO symbol arrange and sign map before, whole LDPC code block is not necessary available.Embodiment 1200 among embodiment 1100, Figure 12 among embodiment 1000 Figure 11 among embodiment 900, Figure 10 among embodiment 1600 and embodiment 1700 and Fig. 9 and the embodiment 1300 among Figure 13 are compared, the latter ( embodiment 900,1000,1100,1200,1300) need be by a storage management position, and causes some stand-by period.Yet, to compare with embodiment 1700 with embodiment 1600, when adopting bit among the embodiment 900,1000,1100,1200,1300 to the intersymbol error technology, performance has significant improvement.Therefore, can obtain significant improvement on the performance with this little cost of certain stand-by period.

Obviously, for each embodiment shown in Figure 16 and Figure 17, the present invention can also adopt other big or small LDPC code block (for example, having different total bits) and have the symbol of other figure place (for example, the label of symbol has the n position, the corresponding columns of n).Under situation about not departing from the scope of the present invention with spirit, for the size of LDPC code block and type and the bit interlace mode to symbol, the designer has very big range of choice.

For other interleaving technique, what for example Figure 16 and 17 was provided interlocks, coding LDPC ₁1210 and LDPC ₂1220 performance exceeds LDPC (b) 1205 general 0.5dB to 0.8dB.

Ensuing 4 width of cloth figure be depicted as 4 kinds different based on GRS abnormal LDPC code and 4 arbitrarily the LDPC sign indicating number (be LDPC (c) (1944,972), LDPC (d) (1944,1296), LDPC (e) (1944,487) and LDPC (f) (1944,1620)) performance schematic diagram relatively.According to these 4 of new method of the present invention design different based in the GRS abnormal LDPC code each, have corresponding low-density parity check (LDPC) matrix, these low-density check matrixes are listed in appendix.

Also provide different from 3 of the GRS abnormal LDPC code extra low-density parity check (LDPC) matrixes in the appendix corresponding to 3.

Figure 18 shows that the schematic diagram of the embodiment of the performance comparison 1800 on communication channel based on GRS abnormal LDPC (1944,973) sign indicating number (1) (shown in the label 1820) and the first coding LDPC (c) (1944,972) (shown in label 1810).

Illustrated among Figure 18 with Block Error Rate contrast signal to noise ratio (or E _b/ N _o) 2 encoding rates of form be 1/2 the performance on the AWGN communication channel of being coded in.As seen from the figure, when Block Error Rate be 1.5 * 10 ^-5The time, should exceed LDPC (c) (1944,972) sign indicating number 1810 about 0.33dB based on the performance of GRS abnormal LDPC (1944,973) sign indicating number (1) 1820.

Figure 19 shows that the schematic diagram of the embodiment of the performance comparison 1900 on communication channel based on GRS abnormal LDPC (1944,1297) sign indicating number (2) (shown in the label 1920) and the second coding LDPC (d) (1944,1296) (shown in label 1910).

Figure 20 shows that the schematic diagram of the embodiment of the performance comparison 2000 on communication channel based on GRS abnormal LDPC (1944,487) sign indicating number (3) (shown in label 2020) and the 3rd coding LDPC (e) (1944,486) (shown in label 2010).

Shown in Figure 21 is schematic diagram based on an embodiment of the performance comparison 2100 of GRS abnormal LDPC (1944,1621) sign indicating number (4) (shown in label 2120) and the 4th coding LDPC (f) (1944,1620) (shown in label 2110).

Figure 22 A is depicted as the schematic diagram of an embodiment of the method 2200 that generates ldpc coded signal.Described method 2200 in step 2210, make up one with based on the corresponding generator matrix of the parity matrix of GRS abnormal LDPC code.For the parity matrix of any one corresponding LDPC sign indicating number, can set up a corresponding generator matrix.Then, in the step 2220, method 2200 uses described generator matrix that at least one information bit is encoded, thereby generates at least one LDPC code word of ldpc coded signal.This encoding operation can be considered in the encoder of the transmitter terminal of communication channel and carries out.After the generator matrix of use setting up (corresponding parity matrix) is encoded at least one information bit, can use this parity matrix that is used to set up described generator matrix that described LDPC coded message is decoded.The present invention can adopt any hardware device (for example, transmitter, transceiver, encoder etc.) of this encoding function that comprises to carry out this operation in the communication system of any kind.

Figure 22 B is depicted as the schematic diagram of an embodiment of the method 2205 that ldpc coded signal is decoded.Described method 2205 receives ldpc coded signal in step 2215.Then, described method 2205 is in step 2225, use corresponding described parity matrix to decode, thereby at least one encoded information bits that described LDPC sign indicating number is compiled in the signal is made best estimation using the ldpc coded signal of encoding based on the GRS abnormal LDPC code based on the GRS abnormal LDPC code.From some angle, this decode operation can be considered at the decoder of the receiver end that is positioned at communication channel to be carried out.The present invention can adopt any hardware device (for example, transmitter, transceiver, decoder or the like) of decoding function that comprises to carry out this operation in various communication system.

In addition, it should be noted that the arrangement of corresponding parity matrix based on the GRS abnormal LDPC code can be adopted following form, wherein P is the permutation matrix of n * n (for example 81 * 81).

Some examples of P matrix following (being depicted as 3 * 3 embodiment)

$P=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),$ Or $P=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$

Null matrix x can be expressed as (shown in following various embodiment):

$X=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$ 3 * 3 embodiment

$x=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$ 4 * 4 embodiment

The embodiment of n * n

Among the embodiment, be in 1944 the LDPC sign indicating number time when being applied to block length, permutation matrix P and null matrix x are 81 * 81 matrix.

The first possible coding structure be with encoding rate be 973/1944 (〉 1/2, be approximately 0.5) based on the corresponding parity check matrix H of GRS abnormal LDPC code for the basis.The form of this parity check matrix H is as follows: H=[H _a, H _b].Because the size of this parity check matrix H, it is represented with two sections.First segment table shows 1 to 12 row and 1 to 12 row, and second segment table shows 13 to 24 row and 1 to 12 row.

$H_a=\left(\begin{array}{cccccccccccc}P& P& P& P& x& x& P& x& x& P& x& x\\ P& P& P& x& P& x& x& P& x& x& P& x\\ P& P& P& x& x& P& x& x& P& x& x& P\\ P& P& P& P& x& x& P& x& x& P& x& x\\ P& P& P& x& P& x& x& P& x& P& x& x\\ P& P& P& x& P& P& x& x& P& x& x& P\\ P& P& P& x& x& P& x& x& P& x& x& P\\ P& P& P& x& P& x& x& P& x& x& P& x\\ P& P& P& x& x& P& x& x& P& x& x& P\\ P& P& P& P& x& x& P& x& x& P& x& x\\ P& P& P& x& P& x& x& P& x& x& P& x\\ P& P& P& x& x& P& x& x& P& x& x& P\end{array}\right)$

$H_b=\left(\begin{array}{cccccccccccc}P& x& x& x& x& x& x& x& x& x& x& P\\ P& P& x& x& x& x& x& x& x& x& x& x\\ x& P& P& x& x& x& x& x& x& x& x& x\\ x& x& P& P& x& x& x& x& x& x& x& x\\ x& x& x& P& P& x& x& x& x& x& x& x\\ x& x& x& x& P& P& x& x& x& x& x& x\\ x& x& x& x& x& P& P& x& x& x& x& x\\ x& x& x& x& x& x& P& P& x& x& x& x\\ x& x& x& x& x& x& x& P& P& x& x& x\\ x& x& x& x& x& x& x& x& P& P& x& x\\ x& x& x& x& x& x& x& x& x& P& P& x\\ x& x& x& x& x& x& x& x& x& x& P& P\end{array}\right)$

The second possible coding structure be based on encoding rate be 2/3 (being approximately 0.667) based on the corresponding parity check matrix H of GRS abnormal LDPC code.The form of this parity check matrix H is as follows: H=[H _a, H _b].Because the size of this parity check matrix H, it is represented with two sections.First segment table shows 1 to 12 row and 1 to 8 row, and second segment table shows 13 to 24 row and 1 to 8 row.

$H_a=\left(\begin{array}{cccccccccccc}P& P& P& P& P& x& P& x& P& x& P& x\\ P& P& P& P& x& P& x& P& x& P& x& P\\ P& P& P& P& P& x& P& x& P& x& P& x\\ P& P& P& P& x& P& x& P& x& P& x& P\\ P& P& P& P& P& x& P& x& P& x& P& x\\ P& P& P& P& x& P& x& P& x& P& x& P\\ P& P& P& P& P& x& P& x& P& x& P& x\\ P& P& P& P& x& P& x& P& x& P& x& P\end{array}\right)$

$H_b=\left(\begin{array}{cccccccccccc}P& x& P& x& P& P& x& x& x& x& x& x\\ x& P& x& P& x& P& P& x& x& x& x& x\\ P& x& P& x& x& x& P& P& x& x& x& x\\ x& P& x& P& x& x& x& P& P& x& x& x\\ P& x& P& x& x& x& x& x& P& P& x& x\\ x& P& x& P& x& x& x& x& x& P& P& x\\ P& x& P& x& x& x& x& x& x& x& P& P\\ x& P& x& P& x& x& x& x& x& x& x& P\end{array}\right)$

The 3rd possible coding structure be based on encoding rate be 3/4 based on the corresponding parity check matrix H of GRS abnormal LDPC code.The form of this parity check matrix H is as follows: H=[H _a, H _b].Because the size of this parity check matrix H, it is represented with two sections.First segment table shows 1 to 12 row and 1 to 6 row, and second segment table shows 13 to 24 row and 1 to 6 row.

$H_a=\left(\begin{array}{cccccccccccc}P& P& P& P& P& P& x& P& P& P& P& x\\ P& P& P& P& P& P& x& x& P& P& P& P\\ P& P& P& P& P& P& P& x& x& P& P& P\\ P& P& P& P& P& P& P& P& x& x& P& P\\ P& P& P& P& P& P& P& P& P& x& x& P\\ P& P& P& P& P& P& P& P& P& P& x& x\end{array}\right)$

$H_a=\left(\begin{array}{cccccccccccc}x& P& P& P& P& x& P& P& x& x& x& x\\ x& x& P& P& P& P& x& P& P& x& x& x\\ P& x& x& P& P& P& x& x& P& P& x& x\\ P& P& x& x& P& P& x& x& x& P& P& x\\ P& P& P& x& x& P& x& x& x& x& P& P\\ P& P& P& P& x& x& x& x& x& x& x& P\end{array}\right)$

The 4th possible coding structure be based on be 5/6 with encoding rate (approximate 0.833) based on the corresponding parity check matrix H of GRS abnormal LDPC code.The form of this parity check matrix H is as follows: H=[H _a, H _b].Because the size of this parity check matrix H, it is represented with two sections.First segment table shows 1 to 12 row and 1 to 4 row, and second segment table shows 13 to 24 row and 1 to 4 row.

$H_a=\left(\begin{array}{cccccccccccc}P& P& P& P& P& P& P& P& P& P& P& P\\ P& P& P& P& P& P& P& P& P& P& P& P\\ P& P& P& P& P& P& P& P& P& P& P& P\\ P& P& P& P& P& P& P& P& P& P& P& P\end{array}\right)$

$H_b=\left(\begin{array}{cccccccccccc}P& P& P& P& P& P& P& P& P& x& x& x\\ P& P& P& P& P& P& P& P& P& P& x& x\\ P& P& P& P& P& P& P& P& x& P& P& x\\ P& P& P& P& P& P& P& P& x& x& P& P\end{array}\right)$

More than among each embodiment, value and the form of H and P are provided as: wherein each permutation matrix P and each null matrix x are 81 * 81 embodiment.

Under various encoding rates and signal to noise ratio, can reach lower Block Error Rate based on the GRS abnormal LDPC code according to what above-mentioned each possible parity matrix was set up.In addition, these codings also have lower wrong rank.Each reduces to 1.5 * 10 based on the emulation Block Error Rate of GRS abnormal LDPC code ^-5, this Block Error Rate is applicable to the aggregated frame of 8192 bytes.

The performance of the described first possible coding structure such as above-mentioned in conjunction with description that Figure 18 carried out.

Shown in Figure 23 is based on the schematic diagram of GRS abnormal LDPC (1944,1296) sign indicating number (5) (shown in label 2320) with an embodiment of the performance comparison 2300 of the second coding LDPC (d) (1944,1296) (shown in label 2310) on communication channel.

Contrast signal to noise ratio (or E with Block Error Rate among Figure 23 _b/ N ₀) form to show these two encoding rates be 2/3 the performance on the AWGN communication channel of being coded in.As seen from the figure, be 1.5 * 10 at Block Error Rate ^-5The time, should exceed LDPC (d) (1944,1296) sign indicating number 2310 about 0.6dB based on the performance of GRS abnormal LDPC (1944,1296) sign indicating number (5) 2320.

Shown in Figure 24 is based on the schematic diagram of GRS abnormal LDPC (1944,486) sign indicating number (6) (shown in label 2420) with an embodiment of the performance comparison 2400 of the 3rd coding LDPC (e) (1944,486) (shown in label 2410) on communication channel.

Figure 24 contrasts signal to noise ratio (or E with Block Error Rate _b/ N ₀) form to show encoding rate be two performances that are coded on the AWGN communication channel of 3/4.As seen from the figure, be 1.5 * 10 at Block Error Rate ^-5The time, should exceed LDPC (e) (1944,486) sign indicating number 2410 about 0.22dB based on the performance of GRS abnormal LDPC (1944,486) sign indicating number (6) 2420.

Shown in Figure 25 is based on the schematic diagram of GRS abnormal LDPC (1944,1620) sign indicating number (7) (shown in label 2520) with an embodiment of the performance comparison 2500 of the 4th coding LDPC (f) (1944,1620) (shown in label 2510) on communication channel.

Contrast signal to noise ratio (or E with Block Error Rate among Figure 25 _b/ N ₀) form to show encoding rate be two performances that are coded on the AWGN communication channel of 5/6.As seen from the figure, be 1.5 * 10 at Block Error Rate ^-5The time, should exceed LDPC (f) (1944,1620) sign indicating number 2510 about 0.11dB based on the performance of GRS abnormal LDPC (1944,1620) sign indicating number (7) 2520.

The complexity of the coding structure that each is possible can reduce the function of the sideline sum of corresponding LDPC two-dimensional plot; This is directly related with the required memory space of information.The poorest situation is 648, than more shown in the following table:

Encoding rate=1/2	LDPC (c) (1944,972), 6966 sidelines	Based on GRS abnormal LDPC (1944,973) sign indicating number (1), 7776 sidelines
			Encoding rate=2/3	LDPC (d) (1944,1296), 7128 sidelines	Based on GRS abnormal LDPC (1944,1296) sign indicating number (5), 7695 sidelines
Encoding rate=3/4	LDPC (e) (1944,486), 6803 sidelines	Based on GRS abnormal LDPC (1944,486) sign indicating number (6), 7695 sidelines

Encoding rate=5/6

LDPC (f) (1944,1620), 6803 sidelines

Based on GRS abnormal LDPC (1944,1620) sign indicating number (7), 7047 sidelines

Table 2

Utilize various methods for designing provided herein, have LDPC sign indicating number than the complete series of known coded more performance and can offer the designer and select for use.Equally, the low-down wrong rank that provide of these codings are highly suitable in the application of high-throughput.In addition, after proposing this improvement in performance, the complexity that is used for decoder that this code signal is decoded is relatively low.

Shown in Figure 26 is first based on GRS abnormal LDPC code (LDPC ₁), second based on GRS abnormal LDPC code (LDPC ₂) using dissimilar bits after the intersymbol error technology with an optional LDPC sign indicating number LDPC (b), the schematic diagram of an embodiment of the performance comparison 2600 on communication channel.This communication channel is a Rayleigh decay communication channel, and the modulator approach of employing is 64QAM.Each coding (LDPC ₁And LDPC ₂) the size of LDPC code block be 1944, the decoding iterations of each performance curve is 12 among the figure.The encoding rate of LDPC sign indicating number LDPC (b) is 2/3 arbitrarily, and code length is 1944, and is abnormal LDPC code.

To should be based on GRS abnormal LDPC code (LDPC ₁) parity matrix can use the submatrix of modification that provides according to above-mentioned equation 11 to make up (, providing this equation once more) for helping reader:

$H\left(1\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^1,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 11)

Should be based on GRS abnormal LDPC code (LDPC ₁) encoding rate be 0.667.

To should be based on GRS abnormal LDPC code (LDPC ₂) parity matrix can use the submatrix of modification that provides according to above-mentioned equation 12 to make up (, providing this equation once more) for helping reader:

$H\left(3\right)=[{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="C200610005277E01731.png,C200610005277E01741.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="C200610005277E01711.png,C200610005277E01731.png"\; state="\{\{state\}\}">H</figure-callout>}}_3^1]$ (equation 12)

Should be based on GRS abnormal LDPC code (LDPC ₂) encoding rate be similarly 0.667.

The bit that can adopt several different types is to the intersymbol error technology; Some of them are staggered respectively shown in Fig. 9 (п 2), Figure 10 (п 3), Figure 11 (п 4), Figure 12 (п 5), Figure 13 (п 6), Figure 16 (п 0) and Figure 17 (п 1).Obviously, the present invention can adopt optional column permutation and not depart from the scope of the present invention and spirit.

Specifically, based on GRS abnormal LDPC code (LDPC ₁) performance use bit to represent to intersymbol error (п 2) (shown in label 2612) to intersymbol error (п 1) (shown in label 2611) and bit.

Based on GRS abnormal LDPC code (LDPC ₂) performance use bit to intersymbol error (п 1) (shown in label 2621), bit to intersymbol error (п 2) (shown in label 2622), bit to intersymbol error

(п 4) (shown in label 2624), bit are represented to intersymbol error (п 6) (shown in label 2626) to intersymbol error (п 5) (shown in label 2625), bit.

The performance of optional LDPC sign indicating number LDPC (b) use bit to intersymbol error (п 0) (shown in label 2630), bit to intersymbol error (п 1) (shown in label 2631), bit to intersymbol error (п 2)

(shown in label 2632), bit are represented to intersymbol error (п 6) (shown in label 2636) to intersymbol error (п 3) (shown in label 2633), bit.

This shows, be 1.5 * 10 at Block Error Rate ^-5The time, each is based on GRS abnormal LDPC code (LDPC ₁) and based on GRS abnormal LDPC code (LDPC ₂) performance exceed approximately 0.8dB of this optional LDPC sign indicating number LDPC (b).

The above various performance map of contrast encode mutually based on GRS abnormal LDPC code and other according to what method provided by the invention was set up that specific energy provides more performance as can be known, can provide lower wrong rank based on the Block Error Rate as the function of signal to noise ratio.

In addition, clearly, select suitable bit can provide more obvious improvement in performance for the various LDPC codings that compare herein to intersymbol error.This principle can extend to the various dissimilar LDPC sign indicating number of not mentioning in this application.The bit of LDPC code block can use a lot of methods to realize to intersymbol error.

A kind of possible method is: will have higher coding intensity (coding strength) in the LDPC code block (promptly, higher bit degree, thereby be connected with more relatively sideline between node on the throne and the corresponding check node) the LDPC bits of coded corresponding with the least significant bit of n position label, described n position label is according to a kind of modulator approach mapped symbol (having planisphere and corresponding mapping).Described method also will have low coding intensity (promptly in the LDPC code block, lower bit degree, thereby be connected with less relatively sideline between node on the throne and the corresponding check node) the LDPC bits of coded corresponding with the highest significant position of n position label, described n position label is according to a kind of modulator approach mapped symbol (having planisphere and corresponding mapping).Described method with the relative more weak bit address in those LDPC bits of coded (in the LDPC code block) with higher relatively coding intensity and the described n position label (promptly, least significant bit) alignment, and those LDPC bits of coded (in the LDPC code block) with relatively low coding intensity are alignd with the relative stronger bit address (that is highest significant position) in the label of described n position.The method is called " by force to weak and weak to strong ".

Another kind of possible method is: will have higher coding intensity in the LDPC code block (promptly, higher bit degree, thereby be connected with more relatively sideline between node on the throne and the corresponding check node) the LDPC bits of coded corresponding with the highest significant position of n position label, described n position label is according to a kind of modulator approach mapped symbol (having planisphere and corresponding mapping).Described method also will have low coding intensity (promptly in the LDPC code block, lower bit degree, thereby be connected with less relatively sideline between node on the throne and the corresponding check node) the LDPC bits of coded corresponding with the least significant bit of n position label, described n position label is according to a kind of modulator approach mapped symbol (having planisphere and corresponding mapping).Described method with the relative stronger bit address in those LDPC bits of coded (in the LDPC code block) with higher relatively coding intensity and the described n position label (promptly, highest significant position) alignment, and those LDPC bits of coded (in the LDPC code block) with relatively low coding intensity are alignd with the relative more weak bit address (that is least significant bit) in the label of described n position.The method is called " by force to strong and weak to weak ".

Equally, after considering above-mentioned 6 labels and interlocking, in I, Q (homophase, quadrature) realized, preceding 3 (being begun by highest significant position) can be considered same phase constituent, and back 3 (till least significant bits) can be considered orthogonal component.Hence one can see that, and it is staggered that each 3 hyte also can be suitable, and like this, these 3 hytes also can be shone upon according to above-mentioned " by force to weak and weak to strong " and " by force to by force with weak to weak " method.

For example, same phase constituent (highest significant position and 2 afterwards) and orthogonal component (least significant bit and before 2) all can shine upon according to " by force to weak and weak to strong " method.Equally, same phase constituent (highest significant position and 2 afterwards) and orthogonal component (least significant bit and before 2) also can shine upon according to " by force to strong and weak to weak " method.

Even in other method, same phase constituent (highest significant position and afterwards 2) can shine upon according to " by force to weak and weak to strong " method, and orthogonal component (least significant bit and before 2) can shine upon according to " by force to strong and weak to weak " method.In another approach, same phase constituent (highest significant position and afterwards 2) can shine upon according to " by force to strong and weak to weak " method, and orthogonal component (least significant bit and before 2) can shine upon according to " by force to weak and weak to by force " method.

Yet after having considered above-mentioned various interleaving techniques and above-mentioned various staggered combination, clearly, it is not have general conclusion that any bit can provide optimum performance to the intersymbol error method.For example, after above-mentioned when considering " by force to weak and weak to strong " or " by force to by force with weak " and the various combination thereof to weak (for example, when independent consideration during) with phase constituent and orthogonal component, clearly, there is not the universal method that to stipulate to select the intensity of LDPC bits of coded is mapped to and will become the bit address of the n position label of mapping symbols.

More appropriate says, optimum performance is the coding selected (for example, selection based on the GRS abnormal LDPC code) and the bit the selected function to intersymbol error.After selecting specific LDPC sign indicating number, obviously can consider multiple bit to intersymbol error to find out a kind of combination that optimum performance can be provided.

The present invention of it should be noted that aforementioned introduction can also be applied to multiple suitable system and/or device be provided with in and do not depart from the scope of the present invention and spirit.

In conjunction with the accompanying drawings to the detailed introduction of the specific embodiment of the invention, other modifications and changes are conspicuous, and these modifications and changes do not break away from spirit of the present invention and scope according to above.

Appendix:

Below provide corresponding four independently based on four of GRS abnormal LDPC code parity matrixs independently.

The form of each parity matrix is as follows:

This form allows to carry out the message-passing decoder of layering.

Each permutation matrix P wherein _{I, j}Being 81 * 81 matrix, can be null matrix (being that all elements value is zero matrix), or the permutation matrix of unit matrix form.The weight of each row is 2 in last (n-k) row of parity check matrix H.

In addition, each this transposed matrix is the column permutation matrix of the original matrix set up by 81 * 81 submatrixs (or permutation matrix), thereby as long as this matrix is carried out line operate, just can set up matrix [H ', T], and wherein T is a upper triangular matrix.[H ', T] can be used for the coding that basis is carried out based on the GRS abnormal LDPC code.The parity matrix that below provides can be used for decoding.

For different each parity matrix of following correspondence based on the GRS abnormal LDPC code, preceding 2 (first row in) quantity of bit node is (promptly in the corresponding LDPC two-dimensional plot of expression respectively, the quantity that is listed as in this parity matrix) and the quantity of check-node (that is, in this parity matrix row quantity).

The next item down represent every row in this parity matrix the verification degree (shown in all the row have identical verification degree so that reduce the complexity of employed decoder; Yet according to the requirement in the special applications, each verification degree can be different values), this generally takes many lines out of.

The position of nonzero term in ensuing this parity matrix of each line display.

Below be an example of the present invention in the 1st:

This parity matrix comprises 1944 row and 972 row.

The verification degree of each row is all 8.

In first row of this parity matrix, nonzero term is positioned at the 0th, 81,162,243,486,729,951 and 1863 row.All other is null value in this first row.

1. based on GRS abnormal LDPC (1944,973) sign indicating number (1), R=0.5

1944 972

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

0 81 162 243 486 729 951 1863

14 102 209 303 494 783 1015 1865

15 103 210 304 495 784 1016 1866

16 104 211 305 496 785 969 1867

17 105 212 306 497 786 1017 1868

18 106 213 307 498 787 1018 1869

19 107 214 308 499 788 1019 1870

20 108 215 309 500 789 970 1871

21 109 216 310 501 790 971 1872

22 110 217 311 502 791 1020 1873

23 111 218 312 503 792 1021 1874

24 112 219 313 504 793 1022 1875

25 113 220 314 505 794 1023 1876

26 114 221 315 506 795 1024 1877

27 115 222 316 507 796 1025 1878

28 116 223 317 508 797 1026 1879

29 117 224 318 509 798 1027 1880

30 118 225 319 510 799 1028 1881

31 119 226 320 511 800 1029 1882

32 120 227 321 512 801 1030 1883

33 121 228 322 513 802 1031 1884

34 122 229 323 514 803 1032 1885

35 123 230 244 515 804 972 1886

36 124 231 245 516 805 1033 1887

37 125 232 246 517 806 1034 1888

38 126 233 247 518 807 1035 1889

39 127 234 248 519 973 1036 1890

40 128 235 249 520 974 1037 1891

41 129 236 250 521 730 1038 1892

42 130 237 251 522 731 1039 1893

43 131 238 252 523 732 1040 1894

44 132 239 253 524 733 1041 1895

45 133 240 254 525 734 1042 1896

46 134 241 255 526 735 1043 1897

47 135 242 256 527 736 1044 1898

48 136 163 257 528 737 1045 1899

49 137 164 258 529 738 1046 1900

50 138 165 259 530 739 1047 1901

51 139 166 260 531 740 1048 1902

52 140 167 261 532 741 1049 1903

53 141 168 262 533 742 1050 1904

54 142 169 263 534 743 1051 1905

55 143 170 264 535 744 1052 1906

56 144 171 265 536 745 952 1907

57 145 172 266 537 746 954 1908

58 146 173 267 538 747 955 1909

59 147 174 268 539 748 957 1910

60 148 175 269 540 749 960 1911

61 149 176 270 541 750 944 1912

62 150 177 271 542 751 945 1913

63 151 178 272 543 752 946 1914

64 152 179 273 544 753 947 1915

65 153 180 274 545 754 950 1916

66 154 181 275 546 755 953 1917

67 155 182 276 547 756 956 1918

68 156 183 277 548 757 999 1919

69 157 184 278 549 758 958 1920

70 158 185 279 550 759 959 1921

71 159 186 280 551 760 961 1922

72 160 187 281 552 761 1000 1923

73 161 188 282 553 762 1001 1924

74 82 189 283 554 763 1002 1925

75 83 190 284 555 764 963 1926

76 84 191 285 556 765 964 1927

77 85 192 286 557 766 962 1928

78 86 193 287 558 767 965 1929

79 87 194 288 559 768 1003 1930

80 88 195 289 560 769 1004 1931

1 89 196 290 561 770 1005 1932

2 90 197 291 562 771 1006 1933

3 91 198 292 563 772 1007 1934

4 92 199 293 564 773 1008 1935

5 93 200 294 565 774 966 1936

6 94 201 295 566 775 1009 1937

7 95 202 296 487 776 1010 1938

8 96 203 297 488 777 1011 1939

9 97 204 298 489 778 1012 1940

10 98 205 299 490 779 1013 1941

11 99 206 300 491 780 967 1942

12 100 207 301 492 781 1014 1943

13 101 208 302 493 782 968 1864

15 148 220 394 616 837 957 1063

67 113 241 331 574 873 1037 1090

55 129 200 337 601 857 969 1084

68 105 208 348 578 843 1026 1123

39 117 191 385 583 840 1002 1131

14 83 238 345 621 881 1020 1114

28 111 206 349 644 845 951 1081

9 81 209 364 643 835 946 1129

7 152 229 381 637 830 1033 1132

40 125 216 360 596 821 961 1072

74 160 242 330 604 870 960 1059

12 149 192 359 587 811 966 1085

5 94 180 375 634 847 959 1115

47 159 193 351 602 866 1039 1103

54 131 164 363 605 875 1001 1116

56 82 219 329 625 822 1016 1087

34 104 233 357 612 879 1003 1062

77 88 214 324 638 884 950 1076

73 154 212 398 588 842 945 1057

17 151 165 371 576 865 1021 1055

71 112 199 326 589 864 956 1088

63 156 217 395 640 858 954 1122

45 146 210 340 615 817 1049 1060

10 141 172 325 629 825 1040 1133

70 132 179 377 610 888 1014 1095

22 101 181 328 608 855 1031 1102

32 122 239 350 641 823 958 1104

38 158 202 334 595 826 1011 1082

49 97 198 400 613 846 1017 1125

6 106 222 397 606 833 1041 1121

46 133 196 358 568 859 971 1065

50 110 188 402 575 889 1024 1119

65 115 170 392 577 877 947 1111

2 153 215 387 635 890 1010 1093

61 96 195 378 598 861 1009 1058

31 95 227 347 594 836 1004 1118

60 89 237 368 618 850 1036 1070

76 128 163 404 592 831 1044 1080

52 136 174 343 584 829 1028 1086

64 119 211 352 646 862 962 1097

30 86 171 379 611 816 1042 1054

58 134 175 356 591 834 1045 1094

0 137 190 361 623 827 999 1098

19 157 207 399 633 869 1052 1113

72 144 186 342 639 876 1005 1130

27 90 236 341 570 878 1029 1109

16 120 185 335 607 856 1019 1079

41 108 201 374 647 819 1030 1108

26 121 177 382 571 815 1007 1124

78 92 189 365 586 839 955 1100

29 147 235 332 603 813 1000 1112

51 161 183 380 582 885 1050 1078

35 142 162 383 632 867 1048 1106

21 140 224 403 581 832 1008 1053

18 93 197 390 597 812 1035 1067

59 127 232 336 573 844 952 1120

23 145 221 366 585 854 1046 1075

13 138 166 354 631 860 967 1064

8 100 231 367 579 871 1018 1089

79 107 203 338 567 828 970 1074

48 109 234 393 620 868 965 1126

69 87 176 327 593 872 1038 1077

25 130 240 388 628 887 1034 1099

44 126 226 386 617 824 944 1083

53 150 223 339 642 883 972 1069

80 124 184 373 627 853 1025 1066

57 116 228 391 599 882 1012 1107

62 98 218 384 630 818 1051 1071

20 143 213 346 572 874 1032 1061

43 123 204 353 636 886 953 1056

42 155 173 355 622 852 964 1127

36 85 194 333 619 880 1006 1096

75 91 230 376 580 810 968 1117

3 102 169 372 624 841 1047 1073

66 139 178 396 614 814 1013 1092

33 99 205 370 609 849 1015 1101

1 103 182 362 600 838 1027 1128

4 118 187 344 569 863 1043 1105

24 135 225 389 590 848 1023 1110

11 114 168 369 626 820 963 1068

37 84 167 401 645 851 1022 1091

16 149 221 425 707 808 1064 1204

38 85 168 478 663 932 1092 1168

68 114 242 449 727 915 1091 1212

56 130 201 424 713 990 1085 1202

69 106 209 438 710 930 1124 1197

40 118 192 419 671 933 1132 1188

15 84 239 417 715 938 1115 1157

29 112 207 450 705 980 1082 1178

10 81 210 484 700 949 1130 1214

8 153 230 422 691 916 1133 1153

41 126 217 415 660 904 1073 1162

75 161 163 457 681 917 1060 1189

13 150 193 464 717 995 1086 1166

6 95 181 466 656 943 1116 1171

48 160 194 444 665 987 1104 1209

55 132 165 407 692 978 1117 1152

57 83 220 483 669 977 1088 1151

35 105 234 427 674 996 1063 1145

78 89 215 481 712 923 1077 1184

74 155 213 473 655 941 1058 1192

18 152 166 455 654 934 1056 1175

72 113 200 420 728 896 1089 1142

64 157 218 480 687 903 1123 1190

46 147 211 432 695 905 1061 1193

11 142 173 442 678 991 1054 1213

71 133 180 448 725 926 1096 1200

23 102 182 459 693 922 1103 1146

33 123 240 416 696 931 1105 1176

39 159 203 456 716 920 1083 1164

50 98 199 460 703 912 1126 1177

7 107 223 475 649 894 1122 1148

47 134 197 412 679 979 1066 1203

51 111 189 471 667 919 1120 1137

66 116 171 441 680 983 1112 1198

3 154 216 470 651 989 1094 1196

62 97 196 406 706 994 1059 1149

32 96 228 462 720 898 1119 1183

61 90 238 474 701 976 1071 1201

77 129 164 440 699 895 1081 1194

53 137 175 468 652 899 1087 1156

65 120 212 405 686 914 1098 1163

31 87 172 429 704 975 1055 1165

59 135 176 482 697 910 1095 1143

0 138 191 437 659 988 1099 1186

20 158 208 426 666 909 1114 1182

73 145 187 451 668 925 1131 1206

28 91 237 436 726 901 1110 1180

17 121 186 408 689 913 1080 1172

42 109 202 439 685 948 1109 1154

27 122 178 461 709 907 1125 1199

79 93 190 445 683 891 1101 1179

30 148 236 431 675 981 1113 1211

52 82 184 428 657 921 1079 1141

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43 83 207 314 554 807 1421 1473

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55 115 193 263 513 741 1397 1470

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36 100 232 300 497 756 1408 1528

65 94 242 292 537 806 1385 1474

1 133 168 274 541 755 1390 1504

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35 91 176 251 552 759 1450 1476

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53 114 239 256 509 774 1402 1507

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51 152 216 346 621 823 1495 1606

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8 159 221 393 641 816 1522 1542

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37 101 233 353 600 842 1529 1594

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2 134 169 330 598 837 1505 1549

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25 83 213 325 639 875 1527 1599

78 150 192 348 645 882 1525 1573

33 96 242 347 576 884 1478 1565

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35 153 241 338 609 819 1492 1614

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41 148 162 389 638 873 1472 1551

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24 99 203 396 603 818 1511 1548

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54 115 240 399 626 874 1508 1562

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68 104 224 390 636 824 1488 1601

26 149 219 352 578 880 1503 1574

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1 97 236 382 586 810 1498 1608

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22 155 227 431 713 936 1598 1638

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38 94 179 254 555 762 1621 1736

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6 100 197 293 513 804 1656 1741

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18 81 218 373 572 844 1735 1812

16 161 238 390 646 839 1734 1855

49 134 225 369 605 830 1728 1851

3 89 171 339 613 879 1767 1795

21 158 201 368 596 820 1775 1849

14 103 189 384 643 856 1758 1841

56 88 202 360 611 875 1725 1823

63 140 173 372 614 884 1773 1788

65 91 228 338 634 831 1776 1848

43 113 242 366 621 888 1716 1800

6 97 223 324 647 813 1703 1810

2 83 221 327 597 851 1729 1816

26 160 174 380 585 874 1759 1827

80 121 208 335 598 873 1747 1784

72 85 226 404 569 867 1760 1824

54 155 219 349 624 826 1731 1828

19 150 181 334 638 834 1706 1843

79 141 188 386 619 817 1720 1860

31 110 190 337 617 864 1781 1839

41 131 168 359 570 832 1779 1809

47 87 211 343 604 835 1732 1838

58 106 207 329 622 855 1766 1854

15 115 231 326 615 842 1704 1830

55 142 205 367 577 868 1777 1842

59 119 197 331 584 818 1739 1808

74 124 179 401 586 886 1746 1836

11 82 224 396 644 819 1748 1782

70 105 204 387 607 870 1726 1797

40 104 236 356 603 845 1769 1850

69 98 166 377 627 859 1765 1805

5 137 172 333 601 840 1709 1794

61 145 183 352 593 838 1763 1819

73 128 220 361 575 871 1755 1804

39 95 180 388 620 825 1737 1856

67 143 184 365 600 843 1702 1807

0 146 199 370 632 836 1762 1829

28 86 216 328 642 878 1714 1813

1 153 195 351 568 885 1724 1799

36 99 165 350 579 887 1730 1796

25 129 194 344 616 865 1741 1837

50 117 210 383 576 828 1778 1801

35 130 186 391 580 824 1738 1791

7 101 198 374 595 848 1742 1786

38 156 164 341 612 822 1757 1857

60 90 192 389 591 814 1774 1826

44 151 162 392 641 876 1753 1847

30 149 233 332 590 841 1723 1803

27 102 206 399 606 821 1752 1822

68 136 241 345 582 853 1768 1831

32 154 230 375 594 863 1744 1858

22 147 175 363 640 869 1756 1835

17 109 240 376 588 880 1722 1840

8 116 212 347 567 837 1750 1798

57 118 163 402 629 877 1701 1821

78 96 185 336 602 881 1711 1820

34 139 169 397 637 816 1764 1814

53 135 235 395 626 833 1719 1853

62 159 232 348 571 812 1708 1861

9 133 193 382 636 862 1733 1844

66 125 237 400 608 811 1718 1811

71 107 227 393 639 827 1770 1859

29 152 222 355 581 883 1721 1862

52 132 213 362 645 815 1743 1802

51 84 182 364 631 861 1727 1789

45 94 203 342 628 889 1713 1815

25 158 230 434 716 938 1794 1863

46 95 204 485 648 911 1816 1865

5 101 240 452 720 978 1846 1866

13 112 179 420 693 915 1834 1867

76 149 188 423 728 920 1847 1868

43 109 215 443 717 940 1818 1869

11 113 192 430 662 901 1793 1870

14 128 197 456 727 900 1807 1871

34 145 235 406 699 894 1788 1872

21 124 178 474 650 933 1786 1873

47 94 177 407 672 941 1819 1874

77 123 171 458 656 924 1853 1875

65 139 210 433 722 998 1791 1876

78 115 218 447 719 979 1784 1877

49 127 201 428 680 942 1826 1878

24 93 168 426 724 990 1833 1879

38 121 216 459 714 982 1835 1880

19 81 219 413 709 895 1813 1881

17 82 239 431 700 925 1856 1882

50 135 226 424 669 913 1852 1883

4 90 172 466 690 926 1796 1884

22 159 202 473 726 897 1850 1885

15 104 190 475 665 984 1842 1886

57 89 203 453 674 949 1824 1887

64 141 174 416 701 935 1789 1888

66 92 229 412 678 809 1849 1889

44 114 163 436 683 898 1801 1890

7 98 224 410 721 932 1811 1891

3 84 222 482 664 936 1817 1892

27 161 175 464 663 943 1828 1893

1 122 209 429 657 905 1785 1894

73 86 227 409 696 912 1825 1895

55 156 220 441 704 914 1829 1896

20 151 182 451 687 892 1844 1897

80 142 189 457 654 976 1861 1898

32 111 191 468 702 975 1840 1899

42 132 169 425 705 986 1810 1900

48 88 212 465 725 929 1839 1901

59 107 208 469 712 921 1855 1902

16 116 232 484 658 903 1831 1903

56 143 206 421 688 981 1843 1904

60 120 198 480 676 928 1809 1905

75 125 180 450 689 988 1837 1906

12 83 225 479 660 997 1782 1907

71 106 205 415 715 896 1798 1908

41 105 237 471 649 907 1851 1909

70 99 167 483 710 808 1806 1910

6 138 173 449 708 904 1795 1911

62 146 184 477 661 908 1820 1912

74 129 221 405 695 923 1805 1913

40 96 181 438 713 980 1857 1914

68 144 185 411 706 919 1808 1915

0 147 200 446 668 996 1830 1916

29 87 217 435 675 918 1814 1917

2 154 196 460 677 934 1800 1918

37 100 166 445 655 910 1797 1919

26 130 195 417 698 922 1838 1920

51 118 211 448 694 995 1802 1921

36 131 187 470 718 916 1792 1922

8 102 199 454 692 891 1787 1923

39 157 165 440 684 987 1858 1924

61 91 193 437 666 930 1827 1925

45 152 162 478 711 993 1848 1926

31 150 234 442 691 985 1804 1927

28 103 207 432 723 899 1823 1928

69 137 242 427 653 992 1832 1929

33 155 231 418 659 977 1859 1930

23 148 176 467 670 994 1836 1931

18 110 241 408 707 909 1841 1932

9 117 213 444 667 893 1799 1933

58 119 164 463 671 948 1822 1934

79 97 186 472 686 939 1821 1935

35 140 170 419 703 917 1815 1936

54 136 236 476 682 989 1854 1937

63 160 233 481 652 983 1862 1938

10 134 194 439 681 931 1845 1939

67 126 238 462 697 937 1812 1940

72 108 228 461 673 906 1860 1941

30 153 223 455 685 927 1783 1942

53 133 214 414 651 991 1803 1943

52 85 183 422 679 902 1790 1864

2. based on GRS abnormal LDPC (1944,1297) sign indicating number (2), R=0.667

1944 647

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

0 81 162 243 324 486 648 810 972 1134 1271 1377

14 102 209 303 338 494 676 886 1011 1177 1324 1447

15 103 210 304 339 495 677 887 1012 1178 1325 1448

16 104 211 305 340 496 678 888 1013 1179 1282 1449

17 105 212 306 341 497 679 889 1014 1180 1281 1450

18 106 213 307 342 498 680 890 1015 1181 1326 1451

19 107 214 308 343 499 681 811 1016 1182 1327 1452

20 108 215 309 344 500 682 812 1017 1183 1284 1453

21 109 216 310 345 501 683 813 1018 1184 1328 1454

22 110 217 311 346 502 684 814 1019 1185 1283 1455

23 111 218 312 347 503 685 815 1020 1186 1285 1456

24 112 219 313 348 504 686 816 1021 1187 1329 1457

25 113 220 314 349 505 687 817 1022 1188 1286 1378

26 114 221 315 350 506 688 818 1023 1189 1287 1379

27 115 222 316 351 507 689 819 1024 1190 1330 1380

28 116 223 317 352 508 690 820 1025 1191 1288 1381

29 117 224 318 353 509 691 821 1026 1192 1331 1382

30 118 225 319 354 510 692 822 1027 1193 1332 1383

31 119 226 320 355 511 693 823 1028 1194 1289 1384

32 120 227 321 356 512 694 824 1029 1195 1333 1385

33 121 228 322 357 513 695 825 1030 1196 1334 1386

34 122 229 323 358 514 696 826 1031 1197 1335 1387

35 123 230 244 359 515 697 827 1032 1198 1290 1388

36 124 231 245 360 516 698 828 1033 1199 1291 1389

37 125 232 246 361 517 699 829 1034 1299 1336 1390

38 126 233 247 362 518 700 830 1035 1201 1292 1391

39 127 234 248 363 519 701 831 1036 1202 1337 1392

40 128 235 249 364 520 702 832 1037 1203 1293 1393

41 129 236 250 365 521 703 833 1038 1204 1338 1394

42 130 237 251 366 522 704 834 1039 1205 1339 1395

43 131 238 252 367 523 705 835 1040 1300 1340 1396

44 132 239 253 368 524 706 836 1041 1207 1294 1397

45 133 240 254 369 525 707 837 1042 1208 1341 1398

46 134 241 255 370 526 708 838 1043 1301 1342 1399

47 135 242 256 371 527 709 839 1044 1210 1343 1400

48 136 163 257 372 528 710 840 1045 1211 1344 1401

49 137 164 258 373 529 711 841 1046 1302 1345 1402

50 138 165 259 374 530 712 842 1047 1303 1295 1403

51 139 166 260 375 531 713 843 1048 1304 1346 1404

52 140 167 261 376 532 714 844 1049 1135 1347 1405

53 141 168 262 377 533 715 845 1050 1136 1348 1406

54 142 169 263 378 534 716 846 1297 1137 1349 1407

55 143 170 264 379 535 717 847 1298 1138 1350 1408

56 144 171 265 380 536 718 848 973 1139 1351 1409

57 145 172 266 381 537 719 849 974 1140 1352 1410

58 146 173 267 382 538 720 850 975 1141 1353 1411

59 147 174 268 383 539 721 851 976 1142 1354 1412

60 148 175 269 384 540 722 852 977 1143 1296 1413

61 149 176 270 385 541 723 853 978 1144 1355 1414

62 150 177 271 386 542 724 854 979 1145 1356 1415

63 151 178 272 387 543 725 855 980 1146 1357 1416

64 152 179 273 388 544 726 856 981 1147 1358 1417

65 153 180 274 389 545 727 857 982 1148 1359 1418

66 154 181 275 390 546 728 858 983 1149 1360 1419

67 155 182 276 391 547 649 859 984 1150 1361 1420

68 156 183 277 392 548 650 860 985 1151 1362 1421

69 157 184 278 393 549 651 861 986 1152 1363 1422

70 158 185 279 394 550 652 862 987 1153 1364 1423

71 159 186 280 395 551 653 863 988 1154 1365 1424

72 160 187 281 396 552 654 864 989 1155 1366 1425

73 161 188 282 397 553 655 865 990 1156 1367 1426

74 821 892 833 985 546 568 669 911 1157 1368 1427

75 831 902 843 995 556 578 679 921 1158 1369 1428

76 84 191 285 400 556 658 868 993 1159 1370 1429

77 85 192 286 401 557 659 869 994 1160 1371 1430

78 86 193 287 402 558 660 870 995 1161 1372 1431

79 87 194 288 403 559 661 871 996 1162 1373 1432

80 88 195 289 404 560 662 872 997 1163 1374 1433

1 89 196 290 325 561 663 873 998 1164 1375 1434

2 90 197 291 326 562 664 874 999 1165 1376 1435

3 91 198 292 327 563 665 875 1000 1166 1273 1436

4 92 199 293 328 564 666 876 1001 1167 1274 1437

5 93 200 294 329 565 667 877 1002 1168 1276 1438

6 94 201 295 330 566 668 878 1003 1169 1277 1439

7 95 202 296 331 487 669 879 1004 1170 1279 1440

8 96 203 297 332 488 670 880 1005 1171 1280 1441

9 97 204 298 333 489 671 881 1006 1172 1323 1442

10 98 205 299 334 490 672 882 1007 1173 1267 1443

11 99 206 300 335 491 673 883 1008 1174 1272 1444

12 100 207 301 336 492 674 884 1009 1175 1275 1445

13 101 208 302 337 493 675 885 1010 1176 1278 1446

15 148 220 319 424 616 785 943 1063 1252 1418 1527

67 113 241 281 448 574 807 914 1090 1234 1453 1459

55 129 200 286 423 601 757 961 1084 1230 1388 1528

68 105 208 244 437 578 745 929 1123 1254 1405 1473

39 117 191 267 418 583 758 932 1131 1228 1384 1538

14 83 238 266 416 621 809 952 1114 1220 1434 1510

28 111 206 260 449 644 784 939 1081 1270 1383 1461

9 81 209 299 483 643 798 965 1129 1247 1399 1483

7 152 229 307 421 637 779 915 1132 1227 1455 1467

40 125 216 290 414 596 777 903 1072 1259 1387 1533

74 160 242 257 456 604 730 916 1059 1200 1433 1530

12 149 192 305 463 587 764 967 1085 1308 1381 1491

5 94 180 308 465 634 782 942 1115 1313 1377 1535

47 159 193 248 443 602 775 956 1103 1243 1422 1525

54 131 164 315 406 605 737 937 1116 1310 1395 1520

56 82 219 261 482 625 744 935 1087 1314 1430 1511

34 104 233 291 426 612 746 968 1062 1222 1419 1480

77 88 214 279 480 638 804 922 1076 1239 1444 1501

73 154 212 292 472 588 767 940 1057 1218 1429 1537

17 151 165 263 454 576 763 933 1055 1052 1401 1476

71 112 199 318 419 589 787 895 1088 1217 1432 1485

63 156 217 252 479 640 761 902 1122 1233 1454 1512

45 146 210 313 431 615 753 904 1060 1316 1438 1489

10 141 172 311 441 629 735 962 1133 1221 1424 1494

70 132 179 264 447 610 780 925 1095 1051 1421 1532

22 101 181 298 458 608 760 921 1102 1322 1382 1475

32 122 239 316 415 641 792 945 1104 1215 1426 1474

38 158 202 309 455 595 802 919 1082 1256 1416 1468

49 97 198 271 459 613 808 911 1125 1229 1411 1507

6 106 222 278 474 606 739 893 1121 1306 1402 1515

46 133 196 280 411 568 776 938 1065 1253 1451 1498

50 110 188 258 470 575 736 918 1119 1309 1392 1465

65 115 170 301 440 577 740 950 1111 1263 1428 1513

2 153 215 297 469 635 755 960 1093 1235 1447 1516

61 96 195 321 485 598 772 966 1058 1266 1456 1536

31 95 227 295 461 594 751 897 1118 1315 1403 1523

60 89 237 287 473 618 801 934 1070 1307 1380 1469

76 128 163 269 439 592 750 894 1080 1258 1385 1499

52 136 174 314 467 584 766 898 1086 1255 1423 1487

64 119 211 294 405 646 742 913 1097 1216 1446 1500

30 86 171 246 428 611 754 930 1054 1260 1445 1471

58 134 175 256 481 591 800 909 1094 1250 1439 1526

0 137 190 262 436 623 748 959 1098 1245 1398 1460

19 157 207 273 425 633 729 908 1113 1236 1406 1521

72 144 186 310 450 639 789 924 1130 1312 1389 1519

27 90 236 270 435 570 762 900 1109 1226 1436 1472

16 120 185 274 407 607 797 912 1079 1262 1404 1506

41 108 201 289 438 647 786 958 1108 1269 1407 1524

26 121 177 306 460 571 731 906 1124 1317 1427 1517

78 92 189 285 444 586 796 891 1100 1237 1414 1479

29 147 235 255 430 603 768 947 1112 1321 1440 1486

51 161 183 284 427 582 799 920 1078 1219 1390 1488

35 142 162 300 468 632 741 955 1106 1257 1378 1466

21 140 224 276 432 581 805 944 1053 1268 1391 1509

18 93 197 288 422 597 791 969 1067 1264 1442 1505

59 127 232 254 417 573 788 954 1120 1212 1417 1529

23 145 221 282 408 585 749 926 1075 1232 1431 1503

13 138 166 243 457 631 793 957 1064 1240 1412 1495

8 100 231 323 478 579 783 899 1089 1223 1410 1477

79 107 203 296 434 567 778 963 1074 1206 1443 1522

48 109 234 251 453 620 769 949 1126 1238 1397 1502

69 87 176 320 462 593 738 946 1077 1241 1415 1534

25 130 240 265 409 628 759 907 1099 1261 1408 1464

44 126 226 250 466 617 795 951 1083 1248 1450 1470

53 150 223 302 471 642 734 941 1069 1213 1457 1481

80 124 184 253 429 627 743 936 1066 1224 1379 1518

57 116 228 275 452 599 770 927 1107 1319 1437 1478

62 98 218 259 451 630 747 896 1071 1225 1400 1482

20 143 213 245 445 572 752 917 1061 1214 1396 1497

43 123 204 322 484 636 790 953 1056 1251 1420 1514

42 155 173 283 412 622 733 892 1127 1265 1394 1493

36 85 194 247 475 619 732 901 1096 1305 1386 1463

75 91 230 317 442 580 806 928 1117 1244 1448 1492

3 102 169 312 410 624 765 905 1073 1246 1413 1508

66 139 178 303 413 614 773 910 1092 1231 1393 1484

33 99 205 272 433 609 756 948 1101 1249 1425 1496

1 103 182 293 420 600 803 971 1128 1242 1435 1462

4 118 187 249 446 569 771 970 1105 1311 1441 1490

24 135 225 268 476 590 774 964 1110 1318 1452 1458

11 114 168 277 464 626 794 923 1068 1320 1409 1531

37 84 167 304 477 645 781 931 1091 1209 1449 1504

16 149 221 320 395 502 707 838 977 1204 1528 1592

38 85 168 305 402 553 663 852 1022 1168 1505 1600

68 114 242 282 332 520 727 874 1038 1302 1460 1590

56 130 201 287 338 488 713 858 1014 1202 1529 1585

69 106 209 245 349 491 710 844 1026 1197 1474 1576

40 118 192 268 386 511 671 841 992 1188 1459 1545

15 84 239 267 346 498 715 882 1020 1157 1511 1566

29 112 207 261 350 524 705 846 972 1178 1462 1602

10 81 210 300 365 554 700 836 981 1304 1484 1541

8 153 230 308 382 542 691 831 1034 1153 1468 1550

41 126 217 291 361 555 660 822 989 1162 1534 1577

75 161 163 258 331 526 681 871 978 1189 1531 1554

13 150 193 306 360 501 717 812 1003 1166 1492 1559

6 95 181 309 376 515 656 848 988 1171 1536 1597

48 160 194 249 352 496 665 867 1040 1301 1526 1540

55 132 165 316 364 494 692 876 991 1152 1521 1619

57 83 220 262 330 527 669 823 1013 1151 1512 1613

35 105 234 292 358 561 674 880 997 1145 1481 1572

78 89 215 280 324 499 712 885 983 1184 1502 1580

74 155 213 293 399 492 655 843 980 1192 1538 1563

18 152 166 264 372 534 654 866 1021 1175 1477 1610

72 113 200 319 327 541 728 865 985 1142 1486 1578

64 157 218 253 396 543 687 859 975 1190 1513 1581

46 147 211 314 341 521 695 818 1050 1193 1490 1601

11 142 173 312 326 564 678 826 1041 1303 1495 1588

71 133 180 265 378 560 725 889 1010 1299 1533 1614

23 102 182 299 329 504 693 856 1031 1146 1476 1564

33 123 240 317 351 558 696 824 987 1176 1475 1552

39 159 203 310 335 550 716 827 1006 1164 1469 1565

50 98 199 272 401 532 703 847 1015 1177 1508 1616

7 107 223 279 398 497 649 834 1042 1148 1516 1591

47 134 197 281 359 557 679 860 1019 1203 1499 1605

51 111 189 259 403 509 667 890 1024 1137 1466 1586

66 116 171 302 393 519 680 878 982 1198 1514 1584

3 154 216 298 388 525 651 811 1005 1196 1517 1617

62 97 196 322 379 536 706 862 1004 1149 1537 1571

32 96 228 296 348 493 720 837 998 1183 1524 1589

61 90 238 288 369 533 701 851 1037 1201 1470 1582

77 129 164 270 325 537 699 832 1045 1194 1500 1544

53 137 175 315 344 552 652 830 1028 1156 1488 1551

65 120 212 295 353 489 686 863 995 1163 1501 1553

31 87 172 247 380 548 704 817 1043 1165 1472 1611

59 135 176 257 357 518 697 835 1046 1143 1527 1574

0 138 191 263 362 547 659 828 986 1186 1461 1570

20 158 208 274 400 563 666 870 973 1182 1522 1594

73 145 187 311 343 539 668 877 999 1300 1520 1568

28 91 237 271 342 551 726 879 1029 1180 1473 1560

17 121 186 275 336 517 689 857 1017 1172 1507 1542

42 109 202 290 375 545 685 820 1030 1154 1525 1587

27 122 178 307 383 486 709 816 1001 1199 1518 1567

79 93 190 286 366 506 683 840 976 1179 1480 1599

30 148 236 256 333 559 675 814 990 1211 1487 1609

52 82 184 285 381 514 657 886 1297 1141 1489 1615

36 143 162 301 384 503 702 868 1049 1147 1467 1546

22 141 225 277 404 528 682 833 1002 1158 1510 1583

19 94 198 289 391 513 714 813 1036 1195 1506 1543

60 128 233 255 337 565 724 845 974 1155 1530 1547

24 146 222 283 367 516 650 855 1047 1159 1504 1562

14 139 167 243 355 538 661 861 1009 1174 1496 1579

9 101 232 244 368 522 698 872 1016 1191 1478 1558

80 108 204 297 339 508 658 829 1018 1170 1523 1608

49 110 235 252 394 505 662 869 996 1140 1503 1557

70 88 177 321 328 546 677 873 1039 1169 1535 1573

26 131 241 266 389 510 694 888 1035 1185 1465 1549

45 127 227 251 387 500 673 825 979 1161 1471 1561

54 151 224 303 340 495 723 884 1033 1173 1482 1607

1 125 185 254 374 566 672 854 1025 1139 1519 1555

58 117 229 276 392 535 688 883 1007 1167 1479 1539

63 99 219 260 385 556 664 819 1298 1134 1483 1596

21 144 214 246 347 512 676 875 1032 1208 1498 1569

44 124 205 323 354 531 722 887 984 1181 1515 1604

43 156 174 284 356 540 670 853 994 1136 1494 1593

37 86 195 248 334 487 648 881 1000 1205 1464 1618

76 92 231 318 377 544 711 810 1011 1150 1493 1603

4 103 170 313 373 549 684 842 1048 1135 1509 1575

67 140 179 304 397 507 719 815 1008 1187 1485 1606

34 100 206 273 371 530 708 850 1012 1138 1497 1548

2 104 183 294 363 529 653 839 1027 1160 1463 1612

5 119 188 250 345 523 718 864 1044 1144 1491 1598

25 136 226 269 390 562 690 849 1023 1210 1458 1595

12 115 169 278 370 490 721 821 993 1207 1532 1556

17 150 222 321 426 618 787 945 1065 1254 1593 1633

13 116 170 279 466 628 796 925 1070 1322 1557 1641

39 86 169 306 479 647 783 933 1093 1212 1601 1631

69 115 163 283 450 576 809 916 1092 1236 1591 1626

57 131 202 288 425 603 759 963 1086 1232 1586 1697

70 107 210 246 439 580 747 931 1125 1256 1577 1666

41 119 193 269 420 585 760 934 1133 1230 1546 1687

16 85 240 268 418 623 731 954 1116 1222 1567 1643

30 113 208 262 451 646 786 941 1083 1311 1603 1662

11 81 211 301 485 645 800 967 1131 1249 1542 1671

9 154 231 309 423 639 781 917 1054 1229 1551 1698

42 127 218 292 416 598 779 905 1074 1261 1578 1675

76 82 164 259 458 606 732 918 1061 1209 1555 1680

14 151 194 307 465 589 766 969 1087 1246 1560 1638

7 96 182 310 467 636 784 944 1117 1315 1598 1661

49 161 195 250 445 604 777 958 1105 1245 1541 1660

56 133 166 317 408 607 739 939 1118 1312 1540 1654

58 84 221 263 484 627 746 937 1089 1316 1614 1693

36 106 235 293 428 614 748 970 1064 1224 1573 1621

79 90 216 281 482 640 806 924 1078 1241 1581 1684

75 156 214 294 474 590 769 942 1059 1220 1564 1651

19 153 167 265 456 578 765 935 1057 1206 1611 1699

73 114 201 320 421 591 789 897 1090 1219 1579 1622

65 158 219 254 481 642 763 904 1124 1235 1582 1642

47 148 212 315 433 617 755 906 1062 1318 1602 1629

12 143 174 313 443 631 737 964 1055 1223 1589 1655

72 134 181 266 449 612 782 927 1097 1200 1615 1685

24 103 183 300 460 610 762 923 1104 1217 1565 1673

34 124 241 318 417 643 794 947 1106 1215 1553 1686

40 160 204 311 457 597 804 921 1084 1258 1566 1657

51 99 200 273 461 615 730 913 1127 1231 1617 1632

8 108 224 280 476 608 741 895 1123 1266 1592 1646

48 135 198 282 413 570 778 940 1067 1255 1606 1627

52 112 190 260 472 577 738 920 1121 1268 1587 1625

67 117 172 303 442 579 742 952 1113 1265 1585 1658

4 155 217 299 471 637 757 962 1095 1237 1618 1692

63 98 197 323 407 600 774 968 1060 1052 1572 1630

33 97 229 297 463 596 753 899 1120 1317 1590 1623

62 91 239 289 475 620 803 936 1072 1213 1583 1665

78 130 165 271 441 594 752 896 1082 1260 1545 1672

54 138 176 316 469 586 768 900 1088 1257 1552 1674

66 121 213 296 405 568 744 915 1099 1218 1554 1652

32 88 173 248 430 613 756 932 1056 1262 1612 1695

60 136 177 258 483 593 802 911 1096 1252 1575 1691

0 139 192 264 438 625 750 961 1100 1247 1571 1635

21 159 209 275 427 635 729 910 1115 1238 1595 1689

74 146 188 312 452 641 791 926 1132 1314 1569 1681

29 92 238 272 437 572 764 902 1111 1228 1561 1663

18 122 187 276 409 609 799 914 1081 1306 1543 1628

43 110 203 291 440 569 788 960 1110 1310 1588 1688

28 123 179 308 462 573 733 908 1126 1319 1568 1640

80 94 191 287 446 588 798 891 1102 1239 1600 1650

31 149 237 257 432 605 770 949 1114 1216 1610 1656

53 83 185 286 429 584 801 922 1080 1221 1616 1667

37 144 162 302 470 634 743 957 1108 1259 1547 1624

23 142 226 278 434 583 807 946 1053 1270 1584 1664

20 95 199 290 424 599 793 971 1069 1269 1544 1668

61 129 234 256 419 575 790 956 1122 1308 1548 1683

25 147 223 284 410 587 751 928 1077 1234 1563 1700

15 140 168 243 459 633 795 959 1066 1242 1580 1679

10 102 233 245 480 581 785 901 1091 1225 1559 1649

1 109 205 298 436 567 780 965 1076 1307 1609 1678

50 111 236 253 455 622 771 951 1128 1240 1558 1694

71 89 178 322 464 595 740 948 1079 1243 1574 1670

27 132 242 267 411 630 761 909 1101 1263 1550 1682

46 128 228 252 468 619 797 953 1085 1250 1562 1648

55 152 225 304 473 644 736 943 1071 1214 1608 1676

2 126 186 255 431 629 745 938 1068 1226 1556 1620

59 118 230 277 454 601 772 929 1109 1321 1539 1637

64 100 220 261 453 632 749 898 1073 1227 1597 1690

22 145 215 247 447 574 754 919 1063 1309 1570 1645

45 125 206 244 406 638 792 955 1058 1253 1605 1634

44 157 175 285 414 624 735 894 1129 1051 1594 1659

38 87 196 249 477 621 734 903 1098 1248 1619 1644

77 93 232 319 444 582 808 930 1119 1305 1604 1696

5 104 171 314 412 626 767 907 1075 1264 1576 1647

68 141 180 305 415 616 775 912 1094 1233 1607 1669

35 101 207 274 435 611 758 950 1103 1251 1549 1653

3 105 184 295 422 602 805 893 1130 1244 1613 1639

6 120 189 251 448 571 773 892 1107 1313 1599 1636

26 137 227 270 478 592 776 966 1112 1320 1596 1677

18 151 223 322 397 504 709 840 979 1300 1634 1781

27 138 228 271 392 564 692 851 1025 1302 1678 1734

14 117 171 280 372 492 723 823 995 1301 1642 1755

40 87 170 307 404 555 665 854 1024 1170 1632 1711

70 116 164 284 334 522 649 876 1040 1304 1627 1730

58 132 203 289 340 490 715 860 1016 1204 1698 1739

71 108 211 247 351 493 712 846 1028 1199 1667 1766

42 120 194 270 388 513 673 843 994 1190 1688 1743

17 86 241 269 348 500 717 884 1022 1159 1644 1748

31 114 209 263 352 526 707 848 972 1180 1663 1706

12 81 212 302 367 556 702 838 983 1136 1672 1729

10 155 232 310 384 544 693 833 1036 1155 1699 1728

43 128 219 293 363 557 662 824 991 1164 1676 1722

77 83 165 260 333 528 683 873 980 1191 1681 1761

15 152 195 308 362 503 719 814 1005 1168 1639 1769

8 97 183 311 378 517 658 850 990 1173 1662 1752

50 82 196 251 354 498 667 869 1042 1211 1661 1719

57 134 167 318 366 496 694 878 993 1154 1655 1767

59 85 222 264 332 529 671 825 1015 1153 1694 1770

37 107 236 294 360 563 676 882 999 1147 1622 1710

80 91 217 282 324 501 714 887 985 1186 1685 1777

76 157 215 295 401 494 657 845 982 1194 1652 1723

20 154 168 266 374 536 656 868 1023 1177 1700 1753

74 115 202 321 329 543 650 867 987 1144 1623 1741

66 159 220 255 398 545 689 861 977 1192 1643 1754

48 149 213 316 343 523 697 820 1298 1195 1630 1725

13 144 175 314 328 566 680 828 1043 1135 1656 1780

73 135 182 267 380 562 727 811 1012 1202 1686 1714

25 104 184 301 331 506 695 858 1033 1148 1674 1775

35 125 242 319 353 560 698 826 989 1178 1687 1773

41 161 205 312 337 552 718 829 1008 1166 1658 1726

52 100 201 274 403 534 705 849 1017 1179 1633 1760

9 109 225 281 400 499 651 836 1044 1150 1647 1778

49 136 199 283 361 559 681 862 1021 1205 1628 1771

53 113 191 261 325 511 669 812 1026 1139 1626 1733

68 118 173 304 395 521 682 880 984 1299 1659 1740

5 156 218 300 390 527 653 813 1007 1198 1693 1742

64 99 198 244 381 538 708 864 1006 1151 1631 1720

34 98 230 298 350 495 722 839 1000 1185 1624 1763

63 92 240 290 371 535 703 853 1039 1203 1666 1759

79 131 166 272 327 539 701 834 1047 1196 1673 1703

55 139 177 317 346 554 654 832 1030 1158 1675 1757

67 122 214 297 355 491 688 865 997 1165 1653 1749

33 89 174 249 382 550 706 819 1045 1167 1696 1731

61 137 178 259 359 520 699 837 1048 1145 1692 1776

0 140 193 265 364 549 661 830 988 1188 1636 1756

22 160 210 276 402 565 668 872 975 1184 1690 1708

75 147 189 313 345 541 670 879 1001 1208 1682 1718

30 93 239 273 344 553 728 881 1031 1182 1664 1724

19 123 188 277 338 519 691 859 1019 1174 1629 1735

44 111 204 292 377 547 687 822 1032 1156 1689 1772

29 124 180 309 385 486 711 818 1003 1201 1641 1732

1 95 192 288 368 508 685 842 978 1181 1651 1736

32 150 238 258 335 561 677 816 992 1303 1657 1751

54 84 186 287 383 516 659 888 973 1143 1668 1768

38 145 162 303 386 505 704 870 1297 1149 1625 1747

24 143 227 279 326 530 684 835 1004 1160 1665 1717

21 96 200 291 393 515 716 815 1038 1197 1669 1746

62 130 235 257 339 487 726 847 976 1157 1684 1762

26 148 224 285 369 518 652 857 1049 1161 1621 1738

16 141 169 243 357 540 663 863 1011 1176 1680 1750

11 103 234 246 370 524 700 874 1018 1193 1650 1716

2 110 206 299 341 510 660 831 1020 1172 1679 1744

51 112 237 254 396 507 664 871 998 1142 1695 1701

72 90 179 323 330 548 679 875 1041 1171 1671 1705

28 133 163 268 391 512 696 890 1037 1187 1683 1758

47 129 229 253 389 502 675 827 981 1163 1649 1713

56 153 226 305 342 497 725 886 1035 1175 1677 1702

3 127 187 256 376 488 674 856 1027 1141 1620 1727

60 119 231 278 394 537 690 885 1009 1169 1638 1712

65 101 221 262 387 558 666 821 974 1134 1691 1764

23 146 216 248 349 514 678 877 1034 1210 1646 1715

46 126 207 245 356 533 724 889 986 1183 1635 1737

45 158 176 286 358 542 672 855 996 1138 1660 1721

39 88 197 250 336 489 648 883 1002 1207 1645 1707

78 94 233 320 379 546 713 810 1013 1152 1697 1704

6 105 172 315 375 551 686 844 1050 1137 1648 1745

69 142 181 306 399 509 721 817 1010 1189 1670 1709

36 102 208 275 373 532 710 852 1014 1140 1654 1779

4 106 185 296 365 531 655 841 1029 1162 1640 1774

7 121 190 252 347 525 720 866 1046 1146 1637 1765

19 152 224 323 428 620 789 947 1067 1256 1702 1788

8 122 191 253 450 573 775 894 1109 1315 1766 1801

28 139 229 272 480 594 778 968 1114 1322 1735 1862

15 118 172 281 468 630 798 927 1072 1217 1756 1860

41 88 171 308 481 569 785 935 1095 1308 1712 1813

71 117 165 285 452 578 731 918 1094 1238 1731 1847

59 133 204 290 427 605 761 965 1088 1234 1740 1785

72 109 212 248 441 582 749 933 1127 1258 1767 1858

43 121 195 271 422 587 762 936 1055 1232 1744 1820

18 87 242 270 420 625 733 956 1118 1224 1749 1827

32 115 210 264 453 568 788 943 1085 1313 1707 1829

13 81 213 303 407 647 802 969 1133 1251 1730 1807

11 156 233 311 425 641 783 919 1056 1231 1729 1850

44 129 220 294 418 600 781 907 1076 1263 1723 1846

78 84 166 261 460 608 734 920 1063 1212 1762 1790

16 153 196 309 467 591 768 971 1089 1264 1770 1844

9 98 184 312 469 638 786 946 1119 1317 1753 1836

51 83 197 252 447 606 779 960 1107 1247 1720 1818

58 135 168 319 410 609 741 941 1120 1314 1768 1783

60 86 223 265 406 629 748 939 1091 1318 1771 1843

38 108 237 295 430 616 750 892 1066 1226 1711 1795

1 92 218 283 484 642 808 926 1080 1243 1778 1805

77 158 216 296 476 592 771 944 1061 1222 1724 1811

21 155 169 267 458 580 767 937 1059 1307 1754 1822

75 116 203 322 423 593 791 899 1092 1221 1742 1859

67 160 221 256 483 644 765 906 1126 1237 1755 1819

49 150 214 317 435 619 757 908 1064 1320 1726 1823

14 145 176 315 445 633 739 966 1057 1225 1781 1838

74 136 183 268 451 614 784 929 1099 1209 1715 1855

26 105 185 302 462 612 764 925 1106 1219 1776 1834

36 126 163 320 419 645 796 949 1108 1215 1774 1804

42 82 206 313 459 599 806 923 1086 1260 1727 1833

53 101 202 275 463 617 732 915 1129 1233 1761 1849

10 110 226 282 478 610 743 897 1125 1052 1779 1825

50 137 200 284 415 572 780 942 1069 1257 1772 1837

54 114 192 262 474 579 740 922 1123 1270 1734 1803

69 119 174 305 444 581 744 954 1115 1051 1741 1831

6 157 219 301 473 639 759 964 1097 1239 1743 1782

65 100 199 245 409 602 776 970 1062 1206 1721 1792

35 99 231 299 465 598 755 901 1122 1319 1764 1845

64 93 241 291 477 622 805 938 1074 1214 1760 1800

80 132 167 273 443 596 754 898 1084 1262 1704 1789

56 140 178 318 471 588 770 902 1090 1259 1758 1814

68 123 215 298 405 570 746 917 1101 1220 1750 1799

34 90 175 250 432 615 758 934 1058 1306 1732 1851

62 138 179 260 485 595 804 913 1098 1254 1777 1802

0 141 194 266 440 627 752 963 1102 1249 1757 1824

23 161 211 277 429 637 729 912 1117 1240 1709 1808

76 148 190 314 454 643 793 928 1054 1316 1719 1794

31 94 240 274 439 574 766 904 1113 1230 1725 1791

20 124 189 278 411 611 801 916 1083 1266 1736 1832

45 112 205 293 442 571 790 962 1112 1312 1773 1796

30 125 181 310 464 575 735 910 1128 1321 1733 1786

296 193 289 448 590 800 891 1104 1241 1737 1861

33 151 239 259 434 607 772 951 1116 1218 1752 1852

55 85 187 288 431 586 803 924 1082 1223 1769 1821

39 146 162 304 472 636 745 959 1110 1261 1748 1842

25 144 228 280 436 585 809 948 1053 1311 1718 1798

22 97 201 292 426 601 795 893 1071 1310 1747 1817

63 131 236 258 421 577 792 958 1124 1246 1763 1826

27 149 225 286 412 589 753 930 1079 1236 1739 1853

17 142 170 243 461 635 797 961 1068 1244 1751 1830

12 104 235 247 482 583 787 903 1093 1227 1717 1835

3 111 207 300 438 567 782 967 1078 1213 1745 1793

52 113 238 255 457 624 773 953 1130 1242 1701 1816

73 91 180 244 466 597 742 950 1081 1245 1706 1815

29 134 164 269 413 632 763 911 1103 1265 1759 1809

48 130 230 254 470 621 799 955 1087 1252 1714 1848

57 154 227 306 475 646 738 945 1073 1309 1703 1856

4 128 188 257 433 631 747 940 1070 1228 1728 1839

61 120 232 279 456 603 774 931 1111 1216 1713 1806

66 102 222 263 455 634 751 900 1075 1229 1765 1854

24 147 217 249 449 576 756 921 1065 1268 1716 1857

47 127 208 246 408 640 794 957 1060 1255 1738 1797

46 159 177 287 416 626 737 896 1131 1200 1722 1784

40 89 198 251 479 623 736 905 1100 1250 1708 1810

79 95 234 321 446 584 730 932 1121 1248 1705 1840

7 106 173 316 414 628 769 909 1077 1269 1746 1828

70 143 182 307 417 618 777 914 1096 1235 1710 1841

37 103 209 276 437 613 760 952 1105 1253 1780 1812

5 107 186 297 424 604 807 895 1132 1305 1775 1787

20 153 225 244 399 506 711 842 981 1208 1789 1863

6 108 187 298 367 533 657 843 1031 1164 1788 1865

9 123 192 254 349 527 722 868 1048 1148 1802 1866

29 140 230 273 394 566 694 853 1027 1304 1783 1867

16 119 173 282 374 494 725 825 997 1211 1861 1868

42 89 172 309 326 557 667 856 1026 1172 1814 1869

72 118 166 286 336 524 651 878 1042 1136 1848 1870

60 134 205 291 342 492 717 862 1018 1300 1786 1871

73 110 213 249 353 495 714 848 1030 1201 1859 1872

44 122 196 272 390 515 675 845 996 1192 1821 1873

19 88 163 271 350 502 719 886 1024 1161 1828 1874

33 116 211 265 354 528 709 850 972 1182 1830 1875

14 81 214 304 369 558 704 840 985 1138 1808 1876

12 157 234 312 386 546 695 835 1038 1157 1851 1877

45 130 221 295 365 559 664 826 993 1166 1847 1878

79 85 167 262 335 530 685 875 982 1193 1791 1879

17 154 197 310 364 505 721 816 1007 1170 1845 1880

10 99 185 313 380 519 660 852 992 1175 1837 1881

52 84 198 253 356 500 669 871 1044 1303 1819 1882

59 136 169 320 368 498 696 880 995 1156 1784 1883

61 87 224 266 334 531 673 827 1017 1155 1844 1884

39 109 238 296 362 565 678 884 1001 1149 1796 1885

2 93 219 284 324 503 716 889 987 1188 1806 1886

78 159 217 297 403 496 659 847 984 1196 1812 1887

22 156 170 268 376 538 658 870 1025 1179 1823 1888

76 117 204 323 331 545 652 869 989 1146 1860 1889

68 161 222 257 400 547 691 863 979 1194 1820 1890

50 151 215 318 345 525 699 822 974 1197 1824 1891

15 146 177 316 330 488 682 830 1045 1137 1839 1892

75 137 184 269 382 564 649 813 1014 1204 1856 1893

27 106 186 303 333 508 697 860 1035 1150 1835 1894

37 127 164 321 355 562 700 828 991 1180 1805 1895

43 83 207 314 339 554 720 831 1010 1168 1834 1896

54 102 203 276 325 536 707 851 1019 1181 1850 1897

11 111 227 283 402 501 653 838 1046 1152 1826 1898

51 138 201 285 363 561 683 864 1023 1207 1838 1899

55 115 193 263 327 513 671 814 1028 1141 1804 1900

70 120 175 306 397 523 684 882 986 1202 1832 1901

7 158 220 302 392 529 655 815 1009 1299 1782 1902

66 101 200 246 383 540 710 866 1008 1153 1793 1903

36 100 232 300 352 497 724 841 1002 1187 1846 1904

65 94 242 292 373 537 705 855 1041 1205 1801 1905

1 133 168 274 329 541 703 836 1049 1198 1790 1906

57 141 179 319 348 556 656 834 1032 1160 1815 1907

69 124 216 299 357 493 690 867 999 1167 1800 1908

35 91 176 251 384 552 708 821 1047 1169 1852 1909

63 139 180 261 361 522 701 839 1050 1147 1803 1910

0 142 195 267 366 551 663 832 990 1190 1825 1911

24 82 212 278 404 487 670 874 977 1186 1809 1912

77 149 191 315 347 543 672 881 1003 1210 1795 1913

32 95 241 275 346 555 650 883 1033 1184 1792 1914

21 125 190 279 340 521 693 861 1021 1176 1833 1915

46 113 206 294 379 549 689 824 1034 1158 1797 1916

31 126 182 311 387 486 713 820 1005 1203 1787 1917

3 97 194 290 370 510 687 844 980 1183 1862 1918

34 152 240 260 337 563 679 818 994 1135 1853 1919

56 86 188 289 385 518 661 890 975 1145 1822 1920

40 147 162 305 388 507 706 872 973 1151 1843 1921

26 145 229 281 328 532 686 837 1006 1162 1799 1922

23 98 202 293 395 517 718 817 1040 1199 1818 1923

64 132 237 259 341 489 728 849 978 1159 1827 1924

28 150 226 287 371 520 654 859 1297 1163 1854 1925

18 143 171 243 359 542 665 865 1013 1178 1831 1926

13 105 236 248 372 526 702 876 1020 1195 1836 1927

4 112 208 301 343 512 662 833 1022 1174 1794 1928

53 114 239 256 398 509 666 873 1000 1144 1817 1929

74 92 181 245 332 550 681 877 1043 1173 1816 1930

30 135 165 270 393 514 698 812 1039 1189 1810 1931

49 131 231 255 391 504 677 829 983 1165 1849 1932

58 155 228 307 344 499 727 888 1037 1177 1857 1933

5 129 189 258 378 490 676 858 1029 1143 1840 1934

62 121 233 280 396 539 692 887 1011 1171 1807 1935

67 103 223 264 389 560 668 823 976 1134 1855 1936

25 148 218 250 351 516 680 879 1036 1302 1858 1937

48 128 209 247 358 535 726 811 988 1185 1798 1938

47 160 178 288 360 544 674 857 998 1140 1785 1939

41 90 199 252 338 491 648 885 1004 1301 1811 1940

80 96 235 322 381 548 715 810 1015 1154 1841 1941

8 107 174 317 377 553 688 846 1298 1139 1829 1942

71 144 183 308 401 511 723 819 1012 1191 1842 1943

38 104 210 277 375 534 712 854 1016 1142 1813 1864

21 154 226 245 430 622 791 949 1069 1258 1362 1863

39 105 211 278 439 615 762 954 1107 1255 1283 1865

7 109 188 299 426 606 809 897 1054 1248 1354 1866

10 124 193 255 452 575 777 896 1111 1317 1327 1867

30 141 231 274 482 596 780 970 1116 1217 1285 1868

17 120 174 283 470 632 800 929 1074 1219 1292 1869

43 90 173 310 483 571 787 937 1097 1246 1350 1870

73 119 167 287 454 580 733 920 1096 1240 1335 1871

61 135 206 292 429 607 763 967 1090 1236 1274 1872

74 111 214 250 443 584 751 935 1129 1260 1334 1873

45 123 197 273 424 589 764 938 1057 1234 1345 1874

20 89 164 272 422 627 735 958 1120 1226 1286 1875

34 117 212 266 455 570 790 945 1087 1315 1336 1876

15 81 215 305 409 569 804 971 1055 1253 1273 1877

13 158 235 313 427 643 785 921 1058 1233 1289 1878

46 131 222 296 420 602 783 909 1078 1265 1271 1879

80 86 168 263 462 610 736 922 1065 1308 1366 1880

18 155 198 311 469 593 770 893 1091 1269 1341 1881

11 100 186 314 471 640 788 948 1121 1319 1374 1882

53 85 199 254 449 608 781 962 1109 1249 1363 1883

60 137 170 321 412 611 743 943 1122 1316 1324 1884

62 88 225 267 408 631 750 941 1093 1320 1373 1885

40 110 239 297 432 618 752 894 1068 1228 1346 1886

3 94 220 285 406 644 730 928 1082 1245 1376 1887

79 160 218 298 478 594 773 946 1063 1224 1329 1888

23 157 171 269 460 582 769 939 1061 1213 1280 1889

77 118 205 244 425 595 793 901 1094 1223 1368 1890

69 82 223 258 485 646 767 908 1128 1239 1365 1891

51 152 216 319 437 621 759 910 1066 1322 1333 1892

16 147 178 317 447 635 741 968 1059 1227 1370 1893

76 138 185 270 453 616 786 931 1101 1212 1360 1894

28 107 187 304 464 614 766 927 1108 1221 1355 1895

38 128 165 322 421 647 798 951 1110 1215 1347 1896

44 84 208 315 461 601 808 925 1088 1262 1328 1897

55 103 204 277 465 619 734 917 1131 1235 1339 1898

12 112 228 284 480 612 745 899 1127 1206 1372 1899

52 139 202 286 417 574 782 944 1071 1259 1281 1900

56 116 194 264 476 581 742 924 1125 1311 1287 1901

71 121 176 307 446 583 746 956 1117 1200 1348 1902

8 159 221 303 475 641 761 966 1099 1241 1332 1903

67 102 201 247 411 604 778 892 1064 1307 1290 1904

37 101 233 301 467 600 757 903 1124 1321 1367 1905

66 95 163 293 479 624 807 940 1076 1309 1282 1906

2 134 169 275 445 598 756 900 1086 1306 1325 1907

58 142 180 320 473 590 772 904 1092 1261 1323 1908

70 125 217 300 405 572 748 919 1103 1222 1344 1909

36 92 177 252 434 617 760 936 1060 1266 1351 1910

64 140 181 262 407 597 806 915 1100 1256 1337 1911

0 143 196 268 442 629 754 965 1104 1251 1277 1912

25 83 213 279 431 639 729 914 1119 1242 1349 1913

78 150 192 316 456 645 795 930 1056 1318 1352 1914

33 96 242 276 441 576 768 906 1115 1232 1371 1915

22 126 191 280 413 613 803 918 1085 1052 1358 1916

47 114 207 295 444 573 792 964 1114 1314 1267 1917

32 127 183 312 466 577 737 912 1130 1216 1293 1918

4 98 195 291 450 592 802 891 1106 1243 1288 1919

35 153 241 261 436 609 774 953 1118 1220 1338 1920

57 87 189 290 433 588 805 926 1084 1225 1275 1921

41 148 162 306 474 638 747 961 1112 1263 1361 1922

27 146 230 282 438 587 731 950 1053 1313 1375 1923

24 99 203 294 428 603 797 895 1073 1312 1356 1924

65 133 238 260 423 579 794 960 1126 1264 1296 1925

29 151 227 288 414 591 755 932 1081 1238 1278 1926

19 144 172 243 463 637 799 963 1070 1305 1343 1927

14 106 237 249 484 585 789 905 1095 1229 1359 1928

5 113 209 302 440 567 784 969 1080 1214 1353 1929

54 115 240 257 459 626 775 955 1132 1244 1284 1930

75 93 182 246 468 599 744 952 1083 1247 1330 1931

31 136 166 271 415 634 765 913 1105 1051 1331 1932

50 132 232 256 472 623 801 957 1089 1254 1279 1933

59 156 229 308 477 568 740 947 1075 1268 1295 1934

6 130 190 259 435 633 749 942 1072 1230 1342 1935

63 122 234 281 458 605 776 933 1113 1218 1364 1936

68 104 224 265 457 636 753 902 1077 1231 1294 1937

26 149 219 251 451 578 758 923 1067 1270 1291 1938

49 129 210 248 410 642 796 959 1062 1257 1326 1939

48 161 179 289 418 628 739 898 1133 1209 1357 1940

42 91 200 253 481 625 738 907 1102 1252 1340 1941

1 97 236 323 448 586 732 934 1123 1250 1369 1942

9 108 175 318 416 630 771 911 1079 1310 1276 1943

72 145 184 309 419 620 779 916 1098 1237 1272 1864

3. based on GRS abnormal LDPC (1944,487) sign indicating number (3), R=0.7505

1944 486

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

0 81 162 243 324 405 567 648 729 810 1053 1134 1215 1296 1434 1539

14 102 209 303 338 426 598 676 783 886 1131 1177 1267 1308 1495 1566

15 103 210 304 339 427 599 677 784 887 1132 1178 1268 1309 1496 1567

16 104 211 305 340 428 600 678 785 888 1133 1179 1269 1310 1497 1568

17 105 212 306 341 429 601 679 786 889 1054 1180 1270 1311 1498 1569

18 106 213 307 342 430 602 680 787 890 1055 1181 1271 1312 1499 1570

19 107 214 308 343 431 603 681 788 811 1056 1182 1272 1313 1454 1571

20 108 215 309 344 432 604 682 789 812 1057 1183 1273 1314 1500 1572

21 109 216 310 345 433 605 683 790 813 1058 1184 1274 1315 1455 1573

22 110 217 311 346 434 606 684 791 814 1059 1185 1275 1316 1501 1574

23 111 218 312 347 435 607 685 792 815 1060 1186 1276 1317 1502 1575

24 112 219 313 348 436 608 686 793 816 1061 1187 1277 1318 1456 1576

25 113 220 314 349 437 609 687 794 817 1062 1188 1278 1319 1503 1577

26 114 221 315 350 438 610 688 795 818 1063 1189 1279 1320 1504 1578

27 115 222 316 351 439 611 689 796 819 1064 1190 1280 1321 1505 1579

28 116 223 317 352 440 612 690 797 820 1065 1191 1281 1322 1506 1580

29 117 224 318 353 441 613 691 798 821 1066 1192 1282 1323 1457 1581

30 118 225 319 354 442 614 692 799 822 1067 1193 1283 1324 1507 1582

31 119 226 320 355 443 615 693 800 823 1068 1194 1284 1325 1508 1583

32 120 227 321 356 444 616 694 801 824 1069 1195 1285 1326 1509 1584

33 121 228 322 357 445 617 695 802 825 1070 1196 1286 1327 1510 1585

34 122 229 323 358 446 618 696 803 826 1071 1197 1287 1328 1511 1586

35 123 230 244 359 447 619 697 804 827 1072 1198 1288 1329 1512 1587

36 124 231 245 360 448 620 698 805 828 1073 1199 1289 1330 1513 1588

37 125 232 246 361 449 621 699 806 829 1074 1200 1290 1331 1514 1589

38 126 233 247 362 450 622 700 807 830 1075 1201 1291 1332 1515 1590

39 127 234 248 363 451 623 701 808 831 1076 1202 1292 1333 1516 1591

40 128 235 249 364 452 624 702 809 832 1077 1203 1293 1334 1517 1592

41 129 236 250 365 453 625 703 730 833 1078 1204 1294 1335 1518 1593

42 130 237 251 366 454 626 704 731 834 1079 1205 1295 1336 1519 1594

43 131 238 252 367 455 627 705 732 835 1080 1206 1216 1337 1458 1595

44 132 239 253 368 456 628 706 733 836 1081 1207 1217 1338 1520 1596

45 133 240 254 369 457 629 707 734 837 1082 1208 1218 1339 1521 1597

46 134 241 255 370 458 630 708 735 838 1083 1209 1219 1340 1522 1598

47 135 242 256 371 459 631 709 736 839 1084 1210 1220 1341 1523 1599

48 136 163 257 372 460 632 710 737 840 1085 1211 1221 1342 1524 1600

49 137 164 258 373 461 633 711 738 841 1086 1212 1222 1343 1525 1601

50 138 165 259 374 462 634 712 739 842 1087 1213 1223 1344 1526 1602

51 139 166 260 375 463 635 713 740 843 1088 1214 1224 1345 1527 1603

52 140 167 261 376 464 636 714 741 844 1089 1135 1225 1346 1528 1604

53 141 168 262 377 465 637 715 742 845 1090 1136 1226 1347 1529 1605

54 142 169 263 378 466 638 716 743 846 1091 1137 1227 1348 1530 1606

55 143 170 264 379 467 639 717 744 847 1092 1138 1228 1349 1531 1607

56 144 171 265 380 468 640 718 745 848 1093 1139 1229 1350 1532 1608

57 145 172 266 381 469 641 719 746 849 1094 1140 1230 1351 1533 1609

58 146 173 267 382 470 642 720 747 850 1095 1141 1231 1352 1534 1610

59 147 174 268 383 471 643 721 748 851 1096 1142 1232 1353 1535 1611

60 148 175 269 384 472 644 722 749 852 1097 1143 1233 1354 1536 1612

61 149 176 270 385 473 645 723 750 853 1098 1144 1234 1355 1537 1613

62 150 177 271 386 474 646 724 751 854 1099 1145 1235 1356 1538 1614

63 151 178 272 387 475 647 725 752 855 1100 1146 1236 1357 1435 1615

64 152 179 273 388 476 568 726 753 856 1101 1147 1237 1358 1437 1616

65 153 180 274 389 477 569 727 754 857 1102 1148 1238 1359 1440 1617

66 154 181 275 390 478 570 728 755 858 1103 1149 1239 1360 1441 1618

67 155 182 276 391 479 571 649 756 859 1104 1150 1240 1361 1443 1619

68 156 183 277 392 480 572 650 757 860 1105 1151 1241 1362 1445 1540

69 157 184 278 393 481 573 651 758 861 1106 1152 1242 1363 1447 1541

70 158 185 279 394 482 574 652 759 862 1107 1153 1243 1364 1430 1542

71 159 186 280 395 483 575 653 760 863 1108 1154 1244 1365 1433 1543

72 160 187 281 396 484 576 654 761 864 1109 1155 1245 1366 1436 1544

73 161 188 282 397 485 577 655 762 865 1110 1156 1246 1367 1438 1545

74 82 189 283 398 406 578 656 763 866 1111 1157 1247 1368 1439 1546

75 83 190 284 399 407 579 657 764 867 1112 1158 1248 1459 1442 1547

76 84 191 285 400 408 580 658 765 868 1113 1159 1249 1370 1485 1548

77 85 192 286 401 409 581 659 766 869 1114 1160 1250 1460 1444 1549

78 86 193 287 402 410 582 660 767 870 1115 1161 1251 1461 1446 1550

79 87 194 288 403 411 583 661 768 871 1116 1162 1252 1462 1448 1551

80 88 195 289 404 412 584 662 769 872 1117 1163 1253 1374 1486 1552

1 89 196 290 325 413 585 663 770 873 1118 1164 1254 1463 1487 1553

2 90 197 291 326 414 586 664 771 874 1119 1165 1255 1464 1488 1554

3 91 198 292 327 415 587 665 772 875 1120 1166 1256 1297 1450 1555

4 92 199 293 328 416 588 666 773 876 1121 1167 1257 1298 1451 1556

5 93 200 294 329 417 589 667 774 877 1122 1168 1258 1299 1449 1557

6 94 201 295 330 418 590 668 775 878 1123 1169 1259 1300 1452 1558

7 95 202 296 331 419 591 669 776 879 1124 1170 1260 1301 1489 1559

8 96 203 297 332 420 592 670 777 880 1125 1171 1261 1302 1490 1560

9 97 204 298 333 421 593 671 778 881 1126 1172 1262 1303 1491 1561

10 98 205 299 334 422 594 672 779 882 1127 1173 1263 1304 1492 1562

11 99 206 300 335 423 595 673 780 883 1128 1174 1264 1305 1493 1563

12 100 207 301 336 424 596 674 781 884 1129 1175 1265 1306 1494 1564

13 101 208 302 337 425 597 675 782 885 1130 1176 1266 1307 1453 1565

15 148 220 319 394 424 706 785 837 943 1203 1252 1356 1418 1591 1631

67 113 241 281 331 448 726 807 873 914 1211 1234 1322 1480 1589 1624

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68 105 208 244 348 437 709 745 843 929 1196 1254 1321 1405 1575 1664

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14 83 238 266 345 416 714 809 881 952 1156 1220 1313 1468 1565 1641

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9 81 209 299 364 483 699 798 835 965 1213 1247 1460 1399 1540 1669

7 152 229 307 381 421 690 779 830 915 1152 1227 1319 1482 1549 1696

40 125 216 290 360 414 659 777 821 903 1161 1259 1296 1387 1576 1673

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12 149 192 305 359 463 716 764 811 967 1165 1275 1333 1381 1558 1636

5 94 180 308 375 465 655 782 847 942 1170 1286 1368 1377 1596 1659

47 159 193 248 351 443 664 775 866 956 1208 1243 1357 1422 1619 1658

54 131 164 315 363 406 691 737 875 937 1151 1283 1302 1395 1618 1652

56 82 219 261 329 482 668 744 822 935 1150 1287 1367 1369 1612 1691

34 104 233 291 357 426 673 746 879 968 1144 1222 1339 1419 1571 1699

77 88 214 279 324 480 711 804 884 922 1183 1239 1370 1428 1579 1682

73 154 212 292 398 472 654 767 842 940 1191 1218 1312 1429 1562 1649

17 151 165 263 371 454 653 763 865 933 1174 1268 1464 1401 1609 1697

71 112 199 318 326 419 727 787 864 895 1141 1217 1362 1467 1577 1700

63 156 217 252 395 479 686 761 858 902 1189 1233 1359 1481 1580 1640

45 146 210 313 340 431 694 753 817 904 1192 1289 1320 1373 1600 1627

10 141 172 311 325 441 677 735 825 962 1212 1221 1364 1424 1587 1653

70 132 179 264 377 447 724 780 888 925 1199 1267 1354 1421 1613 1683

22 101 181 298 328 458 692 760 855 921 1145 1295 1349 1382 1563 1671

32 122 239 316 350 415 695 792 823 945 1175 1215 1340 1426 1551 1684

38 158 202 309 334 455 715 802 826 919 1163 1256 1309 1416 1564 1655

49 97 198 271 400 459 702 808 846 911 1176 1229 1330 1411 1615 1630

6 106 222 278 397 474 728 739 833 893 1147 1264 1366 1402 1590 1644

46 133 196 280 358 411 678 776 859 938 1202 1253 1305 1478 1604 1625

50 110 188 258 402 470 666 736 889 918 1136 1278 1314 1392 1585 1623

65 115 170 301 392 440 679 740 877 950 1197 1263 1341 1465 1583 1656

2 153 215 297 387 469 650 755 890 960 1195 1235 1318 1475 1616 1690

61 96 195 321 378 485 705 772 861 966 1148 1266 1323 1483 1570 1628

31 95 227 295 347 461 719 751 836 897 1182 1288 1361 1403 1588 1621

60 89 237 287 368 473 700 801 850 934 1200 1272 1304 1380 1581 1663

76 128 163 269 404 439 698 750 831 894 1193 1258 1303 1385 1543 1670

52 136 174 314 343 467 651 766 829 898 1155 1255 1297 1423 1550 1672

64 119 211 294 352 405 685 742 862 913 1162 1216 1336 1431 1552 1650

30 86 171 246 379 428 703 754 816 930 1164 1260 1344 1474 1610 1693

58 134 175 256 356 481 696 800 834 909 1142 1250 1327 13751573 1689

0 137 190 262 361 436 658 748 827 959 1185 1245 1374 1398 1569 1633

19 157 207 273 399 425 665 729 869 908 1181 1236 1342 1406 1593 1687

72 144 186 310 342 450 667 789 876 924 1205 1285 1345 1389 1567 1679

27 90 236 270 341 435 725 762 878 900 1179 1226 1365 1372 1559 1661

16 120 185 274 335 407 688 797 856 912 1171 1262 1352 1404 1541 1626

41 108 201 289 374 438 684 786 819 958 1153 1281 1298 1407 1586 1686

26 121 177 306 382 460 708 731 815 906 1198 1290 1328 1427 1566 1638

78 92 189 285 365 444 682 796 839 891 1178 1237 1316 1414 1598 1648

29 147 235 255 332 430 674 768 813 947 1210 1294 1329 1471 1608 1654

51 161 183 284 380 427 656 799 885 920 1140 1219 1300 1390 1614 1665

35 142 162 300 383 468 701 741 867 955 1146 1257 1355 1378 1545 1622

21 140 224 276 403 432 681 805 832 944 1157 1280 1459 1391 1582 1662

18 93 197 288 390 422 713 791 812 969 1194 1279 1350 1376 1542 1666

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23 145 221 282 366 408 649 749 854 926 1158 1232 1301 1466 1561 1698

13 138 166 243 354 457 660 793 860 957 1173 1240 1335 1412 1578 1677

8 100 231 323 367 478 697 783 871 899 1190 1223 1353 1410 1557 1647

79 107 203 296 338 434 657 778 828 963 1169 1270 1346 1473 1607 1676

48 109 234 251 393 453 661 769 868 949 1139 1238 1308 1397 1556 1692

69 87 176 320 327 462 676 738 872 946 1168 1241 1315 1415 1572 1668

25 130 240 265 388 409 693 759 887 907 1184 1261 1317 1408 1548 1680

44 126 226 250 386 466 672 795 824 951 1160 1248 1463 1477 1560 1646

53 150 223 302 339 471 722 734 883 941 1172 1274 1338 1484 1606 1674

80 124 184 253 373 429 671 743 853 936 1138 1224 1334 1379 1554 1620

57 116 228 275 391 452 687 770 882 927 1166 1292 1358 1470 1539 1635

62 98 218 259 384 451 663 747 818 896 1134 1225 1332 1400 1595 1688

20 143 213 245 346 445 675 752 874 917 1207 1276 1324 1396 1568 1643

43 123 204 322 353 484 721 790 886 953 1180 1251 1306 1420 1603 1632

42 155 173 283 355 412 669 733 852 892 1135 1265 1351 1394 1592 1657

36 85 194 247 333 475 648 732 880 901 1204 1246 1331 1386 1617 1642

75 91 230 317 376 442 710 806 810 928 1149 1244 1363 1476 1602 1694

3 102 169 312 372 410 683 765 841 905 1214 1277 1462 1413 1574 1645

66 139 178 303 396 413 718 773 814 910 1186 1231 1299 1393 1605 1667

33 99 205 272 370 433 707 756 849 948 1137 1249 1310 1425 1547 1651

1 103 182 293 362 420 652 803 838 971 1159 1242 1347 1469 1611 1637

4 118 187 249 344 446 717 771 863 970 1143 1284 1307 1472 1597 1634

24 135 225 268 389 476 689 774 848 964 1209 1291 1311 1479 1594 1675

11 114 168 277 369 464 720 794 820 923 1206 1293 1326 1409 1555 1639

37 84 167 304 401 477 662 781 851 931 1167 1271 1343 1432 1599 1629

16 149 221 320 395 425 502 786 838 944 977 1253 1357 1419 1632 1779

38 85 168 305 402 478 553 782 852 932 1022 1272 1344 1477 1630 1709

68 114 242 282 332 449 520 808 874 915 1038 1235 1323 1481 1625 1728

56 130 201 287 338 424 488 758 858 962 1014 1231 1462 1389 1696 1737

69 106 209 245 349 438 491 746 844 930 1026 1255 1322 1406 1665 1764

40 118 192 268 386 419 511 759 841 933 992 1229 1338 1385 1686 1741

15 84 239 267 346 417 498 730 882 953 1020 1221 1314 1469 1642 1746

29 112 207 261 350 450 524 785 846 940 972 1283 1326 1384 1661 1704

10 81 210 300 365 484 554 799 836 966 981 1248 1461 1400 1670 1727

8 153 230 308 382 422 542 780 831 916 1034 1228 1320 1483 1697 1726

41 126 217 291 361 415 555 778 822 904 989 1260 1296 1388 1674 1720

75 161 163 258 331 457 526 731 871 917 978 1270 1361 1468 1679 1759

13 150 193 306 360 464 501 765 812 968 1003 1276 1334 1382 1637 1767

6 95 181 309 376 466 515 783 848 943 988 1287 1459 1377 1660 1750

48 160 194 249 352 444 496 776 867 957 1040 1244 1358 1423 1659 1717

55 132 165 316 364 407 494 738 876 938 991 1284 1303 1396 1653 1765

57 83 220 262 330 483 527 745 823 936 1013 1288 1368 1466 1692 1768

35 105 234 292 358 427 561 747 880 969 997 1223 1340 1420 1700 1708

78 89 215 280 324 481 499 805 885 923 983 1240 1460 1474 1683 1775

74 155 213 293 399 473 492 768 843 941 980 1219 1313 1369 1650 1721

18 152 166 264 372 455 534 764 866 934 1021 1269 1297 1402 1698 1751

72 113 200 319 327 420 541 788 865 896 985 1218 1363 1371 1621 1739

64 157 218 253 396 480 543 762 859 903 975 1234 1360 1482 1641 1752

46 147 211 314 341 432 521 754 818 905 1050 1290 1321 1375 1628 1723

11 142 173 312 326 442 564 736 826 963 1041 1222 1365 1425 1654 1778

71 133 180 265 378 448 560 781 889 926 1010 1268 1355 1422 1684 1712

23 102 182 299 329 459 504 761 856 922 1031 1216 1350 1383 1672 1773

33 123 240 317 351 416 558 793 824 946 987 1215 1341 1427 1685 1771

39 159 203 310 335 456 550 803 827 920 1006 1257 1310 1417 1656 1724

50 98 199 272 401 460 532 809 847 912 1015 1230 1331 1412 1631 1758

7 107 223 279 398 475 497 740 834 894 1042 1265 1367 1403 1645 1776

47 134 197 281 359 412 557 777 860 939 1019 1254 1306 1479 1626 1769

51 111 189 259 403 471 509 737 890 919 1024 1279 1315 1393 1624 1731

66 116 171 302 393 441 519 741 878 951 982 1264 1342 1429 1657 1738

3 154 216 298 388 470 525 756 811 961 1005 1236 1319 1476 1691 1740

62 97 196 322 379 406 536 773 862 967 1004 1267 1324 1484 1629 1718

32 96 228 296 348 462 493 752 837 898 998 1289 1362 1404 1622 1761

61 90 238 288 369 474 533 802 851 935 1037 1273 1305 1381 1664 1757

77 129 164 270 325 440 537 751 832 895 1045 1259 1304 1386 1671 1781

53 137 175 315 344 468 552 767 830 899 1028 1256 1298 1424 1673 1755

65 120 212 295 353 405 489 743 863 914 995 1217 1337 1475 1651 1747

31 87 172 247 380 429 548 755 817 931 1043 1261 1345 1431 1694 1729

59 135 176 257 357 482 518 801 835 910 1046 1251 1328 1471 1690 1774

0 138 191 263 362 437 547 749 828 960 986 1246 1463 1399 1634 1754

20 158 208 274 400 426 563 729 870 909 973 1237 1343 1407 1688 1706

73 145 187 311 343 451 539 790 877 925 999 1286 1346 1390 1680 1716

28 91 237 271 342 436 551 763 879 901 1029 1227 1366 1470 1662 1722

17 121 186 275 336 408 517 798 857 913 1017 1263 1353 1405 1627 1733

42 109 202 290 375 439 545 787 820 959 1030 1282 1299 1408 1687 1770

27 122 178 307 383 461 486 732 816 907 1001 1291 1329 1465 1639 1730

79 93 190 286 366 445 506 797 840 891 976 1238 1317 1415 1649 1734

30 148 236 256 333 431 559 769 814 948 990 1295 1330 1472 1655 1749

52 82 184 285 381 428 514 800 886 921 1051 1220 1301 1391 1666 1766

36 143 162 301 384 469 503 742 868 956 1049 1258 1356 1379 1623 1745

22 141 225 277 404 433 528 806 833 945 1002 1281 1370 1392 1663 1715

19 94 198 289 391 423 513 792 813 970 1036 1280 1351 1473 1667 1744

60 128 233 255 337 418 565 789 845 955 974 1274 1349 1418 1682 1760

24 146 222 283 367 409 516 750 855 927 1047 1233 1302 1467 1699 1736

14 139 167 243 355 458 538 794 861 958 1009 1241 1336 1413 1678 1748

9 101 232 244 368 479 522 784 872 900 1016 1224 1354 1411 1648 1714

80 108 204 297 339 435 508 779 829 964 1018 1271 1347 1428 1677 1742

49 110 235 252 394 454 505 770 869 950 996 1239 1309 1398 1693 1701

70 88 177 321 328 463 546 739 873 947 1039 1242 1316 1416 1669 1703

26 131 241 266 389 410 510 760 888 908 1035 1262 1318 1409 1681 1756

45 127 227 251 387 467 500 796 825 952 979 1249 1464 1478 1647 1711

54 151 224 303 340 472 495 735 884 942 1033 1275 1339 1378 1675 1780

1 125 185 254 374 430 566 744 854 937 1025 1225 1335 1380 1620 1725

58 117 229 276 392 453 535 771 883 928 1007 1293 1359 1373 1636 1710

63 99 219 260 385 452 556 748 819 897 1052 1226 1333 1401 1689 1762

21 144 214 246 347 446 512 753 875 918 1032 1277 1325 1397 1644 1713

44 124 205 323 354 485 531 791 887 954 984 1252 1307 1421 1633 1735

43 156 174 284 356 413 540 734 853 893 994 1266 1352 1395 1658 1719

37 86 195 248 334 476 487 733 881 902 1000 1247 1332 1387 1643 1705

76 92 231 318 377 443 544 807 810 929 1011 1245 1364 1432 1695 1702

4 103 170 313 373 411 549 766 842 906 1048 1278 1374 1414 1646 1743

67 140 179 304 397 414 507 774 815 911 1008 1232 1300 1394 1668 1707

34 100 206 273 371 434 530 757 850 949 1012 1250 1311 1426 1652 1777

2 104 183 294 363 421 529 804 839 892 1027 1243 1348 1372 1638 1772

5 119 188 250 345 447 523 772 864 971 1044 1285 1308 1376 1635 1763

25 136 226 269 390 477 562 775 849 965 1023 1292 1312 1480 1676 1732

12 115 169 278 370 465 490 795 821 924 993 1294 1327 1410 1640 1753

17 150 222 321 396 426 503 618 839 945 978 1065 1358 1420 1780 1786

13 116 170 279 371 466 491 628 822 925 994 1070 1328 1411 1754 1858

39 86 169 306 403 479 554 647 853 933 1023 1093 1345 1478 1710 1811

69 115 163 283 333 450 521 576 875 916 1039 1092 1324 1482 1729 1845

57 131 202 288 339 425 489 603 859 963 1015 1086 1374 1390 1738 1783

70 107 210 246 350 439 492 580 845 931 1027 1125 1323 1407 1765 1856

41 119 193 269 387 420 512 585 842 934 993 1133 1339 1386 1742 1818

16 85 240 268 347 418 499 623 883 954 1021 1116 1315 1372 1747 1825

30 113 208 262 351 451 525 646 847 941 972 1083 1327 1385 1705 1827

11 81 211 301 366 485 555 645 837 967 982 1131 1462 1401 1728 1805

9 154 231 309 383 423 543 639 832 917 1035 1054 1321 1484 1727 1848

42 127 218 292 362 416 556 598 823 905 990 1074 1296 1389 1721 1844

76 82 164 259 332 458 527 606 872 918 979 1061 1362 1469 1760 1788

14 151 194 307 361 465 502 589 813 969 1004 1087 1335 1383 1768 1842

7 96 182 310 377 467 516 636 849 944 989 1117 1370 1377 1751 1834

49 161 195 250 353 445 497 604 868 958 1041 1105 1359 1424 1718 1816

56 133 166 317 365 408 495 607 877 939 992 1118 1304 1397 1766 1861

58 84 221 263 331 484 528 627 824 937 1014 1089 1459 1467 1769 1841

36 106 235 293 359 428 562 614 881 970 998 1064 1341 1421 1709 1793

79 90 216 281 324 482 500 640 886 924 984 1078 1461 1431 1776 1803

75 156 214 294 400 474 493 590 844 942 981 1059 1314 1466 1722 1809

19 153 167 265 373 456 535 578 867 935 1022 1057 1298 1403 1752 1820

73 114 201 320 328 421 542 591 866 897 986 1090 1364 1468 1740 1857

65 158 219 254 397 481 544 642 860 904 976 1124 1361 1483 1753 1817

47 148 212 315 342 433 522 617 819 906 1051 1062 1322 1471 1724 1821

12 143 174 313 327 443 565 631 827 964 1042 1055 1366 1426 1779 1836

72 134 181 266 379 449 561 612 890 927 1011 1097 1356 1423 1713 1853

24 103 183 300 330 460 505 610 857 923 1032 1104 1351 1384 1774 1832

34 124 241 318 352 417 559 643 825 947 988 1106 1342 1465 1772 1802

40 160 204 311 336 457 551 597 828 921 1007 1084 1311 1418 1725 1831

51 99 200 273 402 461 533 615 848 913 1016 1127 1332 1413 1759 1847

8 108 224 280 399 476 498 608 835 895 1043 1123 1368 1404 1777 1823

48 135 198 282 360 413 558 570 861 940 1020 1067 1307 1480 1770 1835

52 112 190 260 404 472 510 577 811 920 1025 1121 1316 1394 1732 1801

67 117 172 303 394 442 520 579 879 952 983 1113 1343 1369 1739 1829

4 155 217 299 389 471 526 637 812 962 1006 1095 1320 1432 1741 1782

63 98 197 323 380 407 537 600 863 968 1005 1060 1325 1378 1719 1790

33 97 229 297 349 463 494 596 838 899 999 1120 1363 1405 1762 1843

62 91 239 289 370 475 534 620 852 936 1038 1072 1306 1382 1758 1798

78 130 165 271 326 441 538 594 833 896 1046 1082 1305 1387 1702 1787

54 138 176 316 345 469 553 586 831 900 1029 1088 1299 1425 1756 1812

66 121 213 296 354 405 490 568 864 915 996 1099 1338 1476 1748 1797

32 88 173 248 381 430 549 613 818 932 1044 1056 134 614 751 730 1849

60 136 177 258 358 483 519 593 836 911 1047 1096 1329 1472 1775 1800

0 139 192 264 363 438 548 625 829 961 987 1100 1464 1400 1755 1822

21 159 209 275 401 427 564 635 871 910 974 1115 1344 1408 1707 1806

74 146 188 312 344 452 540 641 878 926 1000 1132 1347 1391 1717 1792

29 92 238 272 343 437 552 572 880 902 1030 1111 1367 1373 1723 1789

18 122 187 276 337 409 518 609 858 914 1018 1081 1354 1406 1734 1830

43 110 203 291 376 440 546 569 821 960 1031 1110 1300 1409 1771 1794

28 123 179 308 384 462 486 573 817 908 1002 1126 1330 1429 1731 1784

80 94 191 287 367 446 507 588 841 891 977 1102 1318 1416 1735 1859

31 149 237 257 334 432 560 605 815 949 991 1114 1331 1376 1750 1850

53 83 185 286 382 429 515 584 887 922 1052 1080 1302 1392 1767 1819

37 144 162 302 385 470 504 634 869 957 1050 1108 1357 1380 1746 1840

23 142 226 278 325 434 529 583 834 946 1003 1053 1460 1393 1716 1796

20 95 199 290 392 424 514 599 814 971 1037 1069 1352 1428 1745 1815

61 129 234 256 338 419 566 575 846 956 975 1122 1350 1419 1761 1824

25 147 223 284 368 410 517 587 856 928 1048 1077 1303 1371 1737 1851

15 140 168 243 356 459 539 633 862 959 1010 1066 1337 1414 1749 1828

10 102 233 245 369 480 523 581 873 901 1017 1091 1355 1412 1715 1833

1 109 205 298 340 436 509 567 830 965 1019 1076 1348 1474 1743 1791

50 111 236 253 395 455 506 622 870 951 997 1128 1310 1399 1701 1814

71 89 178 322 329 464 547 595 874 948 1040 1079 1317 1417 1704 1813

27 132 242 267 390 411 511 630 889 909 1036 1101 1319 1410 1757 1807

46 128 228 252 388 468 501 619 826 953 980 1085 1297 1479 1712 1846

55 152 225 304 341 473 496 644 885 943 1034 1071 1340 1379 1781 1854

2 126 186 255 375 431 487 629 855 938 1026 1068 1336 1381 1726 1837

59 118 230 277 393 454 536 601 884 929 1008 1109 1360 1375 1711 1804

64 100 220 261 386 453 557 632 820 898 973 1073 1334 1402 1763 1852

22 145 215 247 348 447 513 574 876 919 1033 1063 1326 1398 1714 1855

45 125 206 244 355 406 532 638 888 955 985 1058 1308 1422 1736 1795

44 157 175 285 357 414 541 624 854 894 995 1129 1353 1396 1720 1862

38 87 196 249 335 477 488 621 882 903 1001 1098 1333 1388 1706 1808

77 93 232 319 378 444 545 582 810 930 1012 1119 1365 1477 1703 1838

5 104 171 314 374 412 550 626 843 907 1049 1075 1463 1415 1744 1826

68 141 180 305 398 415 508 616 816 912 1009 1094 1301 1395 1708 1839

35 101 207 274 372 435 531 611 851 950 1013 1103 1312 1427 1778 1810

3 105 184 295 364 422 530 602 840 893 1028 1130 1349 1470 1773 1785

6 120 189 251 346 448 524 571 865 892 1045 1107 1309 1473 1764 1799

26 137 227 270 391 478 563 592 850 966 1024 1112 1313 1481 1733 1860

18 151 223 322 397 427 504 619 709 946 979 1066 1206 1421 1787 1863

27 138 228 271 392 479 564 593 692 967 1025 1113 1212 1482 1861 1865

14 117 171 280 372 467 492 629 723 926 995 1071 1209 1412 1859 1866

40 87 170 307 404 480 555 568 665 934 1024 1094 1170 1479 1812 1867

70 116 164 284 334 451 522 577 649 917 1040 1093 1214 1483 1846 1868

58 132 203 289 340 426 490 604 715 964 1016 1087 1204 1391 1784 1869

71 108 211 247 351 440 493 581 712 932 1028 1126 1199 1408 1857 1870

42 120 194 270 388 421 513 586 673 935 994 1054 1190 1387 1819 1871

17 86 241 269 348 419 500 624 717 955 1022 1117 1159 1470 1826 1872

31 114 209 263 352 452 526 647 707 942 972 1084 1180 1386 1828 1873

12 81 212 302 367 406 556 646 702 968 983 1132 1136 1402 1806 1874

10 155 232 310 384 424 544 640 693 918 1036 1055 1155 1378 1849 1875

43 128 219 293 363 417 557 599 662 906 991 1075 1164 1390 1845 1876

77 83 165 260 333 459 528 607 683 919 980 1062 1191 1372 1789 1877

15 152 195 308 362 466 503 590 719 970 1005 1088 1168 1384 1843 1878

8 97 183 311 378 468 517 637 658 945 990 1118 1173 1377 1835 1879

50 82 196 251 354 446 498 605 667 959 1042 1106 1211 1425 1817 1880

57 134 167 318 366 409 496 608 694 940 993 1119 1154 1398 1862 1881

59 85 222 264 332 485 529 628 671 938 1015 1090 1153 1371 1842 1882

37 107 236 294 360 429 563 615 676 971 999 1065 1147 1422 1794 1883

80 91 217 282 324 483 501 641 714 925 985 1079 1186 1475 1804 1884

76 157 215 295 401 475 494 591 657 943 982 1060 1194 1467 1810 1885

20 154 168 266 374 457 536 579 656 936 1023 1058 1177 1404 1821 1886

74 115 202 321 329 422 543 592 650 898 987 1091 1144 1469 1858 1887

66 159 220 255 398 482 545 643 689 905 977 1125 1192 1484 1818 1888

48 149 213 316 343 434 523 618 697 907 1052 1063 1195 1472 1822 1889

13 144 175 314 328 444 566 632 680 965 1043 1056 1135 1427 1837 1890

73 135 182 267 380 450 562 613 727 928 1012 1098 1202 1424 1854 1891

25 104 184 301 331 461 506 611 695 924 1033 1105 1148 1385 1833 1892

35 125 242 319 353 418 560 644 698 948 989 1107 1178 1429 1803 1893

41 161 205 312 337 458 552 598 718 922 1008 1085 1166 1419 1832 1894

52 100 201 274 403 462 534 616 705 914 1017 1128 1179 1414 1848 1895

9 109 225 281 400 477 499 609 651 896 1044 1124 1150 1405 1824 1896

49 136 199 283 361 414 559 571 681 941 1021 1068 1205 1481 1836 1897

53 113 191 261 325 473 511 578 669 921 1026 1122 1139 1395 1802 1898

68 118 173 304 395 443 521 580 682 953 984 1114 1200 1466 1830 1899

5 156 218 300 390 472 527 638 653 963 1007 1096 1198 1477 1782 1900

64 99 198 244 381 408 538 601 708 969 1006 1061 1151 1379 1791 1901

34 98 230 298 350 464 495 597 722 900 1000 1121 1185 1406 1844 1902

63 92 240 290 371 476 535 621 703 937 1039 1073 1203 1383 1799 1903

79 131 166 272 327 442 539 595 701 897 1047 1083 1196 1388 1788 1904

55 139 177 317 346 470 554 587 654 901 1030 1089 1158 1426 1813 1905

67 122 214 297 355 405 491 569 688 916 997 1100 1165 1432 1798 1906

33 89 174 249 382 431 550 614 706 933 1045 1057 1167 1476 1850 1907

61 137 178 259 359 484 520 594 699 912 1048 1097 1145 1376 1801 1908

0 140 193 265 364 439 549 626 661 962 988 1101 1188 1401 1823 1909

22 160 210 276 402 428 565 636 668 911 975 1116 1184 1409 1807 1910

75 147 189 313 345 453 541 642 670 927 1001 1133 1208 1392 1793 1911

30 93 239 273 344 438 553 573 728 903 1031 1112 1182 1375 1790 1912

19 123 188 277 338 410 519 610 691 915 1019 1082 1174 1407 1831 1913

44 111 204 292 377 441 547 570 687 961 1032 1111 1156 1410 1795 1914

29 124 180 309 385 463 486 574 711 909 1003 1127 1201 1369 1785 1915

1 95 192 288 368 447 508 589 685 891 978 1103 1181 1417 1860 1916

32 150 238 258 335 433 561 606 677 950 992 1115 1213 1473 1851 1917

54 84 186 287 383 430 516 585 659 923 973 1081 1143 1393 1820 1918

38 145 162 303 386 471 505 635 704 958 1051 1109 1149 1381 1841 1919

24 143 227 279 326 435 530 584 684 947 1004 1053 1160 1394 1797 1920

21 96 200 291 393 425 515 600 716 892 1038 1070 1197 1474 1816 1921

62 130 235 257 339 420 487 576 726 957 976 1123 1157 1420 1825 1922

26 148 224 285 369 411 518 588 652 929 1049 1078 1161 1468 1852 1923

16 141 169 243 357 460 540 634 663 960 1011 1067 1176 1415 1829 1924

11 103 234 246 370 481 524 582 700 902 1018 1092 1193 1413 1834 1925

2 110 206 299 341 437 510 567 660 966 1020 1077 1172 1431 1792 1926

51 112 237 254 396 456 507 623 664 952 998 1129 1142 1400 1815 1927

72 90 179 323 330 465 548 596 679 949 1041 1080 1171 1418 1814 1928

28 133 163 268 391 412 512 631 696 910 1037 1102 1187 1411 1808 1929

47 129 229 253 389 469 502 620 675 954 981 1086 1163 1480 1847 1930

56 153 226 305 342 474 497 645 725 944 1035 1072 1175 1380 1855 1931

3 127 187 256 376 432 488 630 674 939 1027 1069 1141 1382 1838 1932

60 119 231 278 394 455 537 602 690 930 1009 1110 1169 1471 1805 1933

65 101 221 262 387 454 558 633 666 899 974 1074 1134 1403 1853 1934

23 146 216 248 349 448 514 575 678 920 1034 1064 1210 1399 1856 1935

46 126 207 245 356 407 533 639 724 956 986 1059 1183 1423 1796 1936

45 158 176 286 358 415 542 625 672 895 996 1130 1138 1397 1783 1937

39 88 197 250 336 478 489 622 648 904 1002 1099 1207 1389 1809 1938

78 94 233 320 379 445 546 583 713 931 1013 1120 1152 1478 1839 1939

6 105 172 315 375 413 551 627 686 908 1050 1076 1137 1416 1827 1940

69 142 181 306 399 416 509 617 721 913 1010 1095 1189 1396 1840 1941

36 102 208 275 373 436 532 612 710 951 1014 1104 1140 1465 1811 1942

4 106 185 296 365 423 531 603 655 894 1029 1131 1162 1373 1786 1943

7 121 190 252 347 449 525 572 720 893 1046 1108 1146 1428 1800 1864

19 152 224 323 398 428 505 620 710 789 980 1067 1207 1256 1531 1863

8 122 191 253 348 450 526 573 721 775 1047 1109 1147 1288 1499 1865

28 139 229 272 393 480 565 594 693 778 1026 1114 1213 1295 1434 1866

15 118 172 281 373 468 493 630 724 798 996 1072 1210 1217 1535 1867

41 88 171 308 325 481 556 569 666 785 1025 1095 1171 1275 1509 1868

71 117 165 285 335 452 523 578 650 731 1041 1094 1135 1238 1443 1869

59 133 204 290 341 427 491 605 716 761 1017 1088 1205 1234 1532 1870

72 109 212 248 352 441 494 582 713 749 1029 1127 1200 1258 1487 1871

43 121 195 271 389 422 514 587 674 762 995 1055 1191 1232 1441 1872

18 87 242 270 349 420 501 625 718 733 1023 1118 1160 1224 1515 1873

32 115 210 264 353 453 527 568 708 788 972 1085 1181 1286 1447 1874

13 81 213 303 368 407 557 647 703 802 984 1133 1137 1251 1493 1875

11 156 233 311 385 425 545 641 694 783 1037 1056 1156 1231 1442 1876

44 129 220 294 364 418 558 600 663 781 992 1076 1165 1263 1537 1877

78 84 166 261 334 460 529 608 684 734 981 1063 1192 1273 1534 1878

16 153 196 309 363 467 504 591 720 768 1006 1089 1169 1279 1454 1879

9 98 184 312 379 469 518 638 659 786 991 1119 1174 1290 1435 1880

51 83 197 252 355 447 499 606 668 779 1043 1107 1212 1247 1529 1881

58 135 168 319 367 410 497 609 695 741 994 1120 1155 1287 1524 1882

60 86 223 265 333 406 530 629 672 748 1016 1091 1154 1291 1516 1883

38 108 237 295 361 430 564 616 677 750 1000 1066 1148 1226 1490 1884

1 92 218 283 324 484 502 642 715 808 986 1080 1187 1243 1457 1885

77 158 216 296 402 476 495 592 658 771 983 1061 1195 1222 1440 1886

21 155 169 267 375 458 537 580 657 767 1024 1059 1178 1272 1451 1887

75 116 203 322 330 423 544 593 651 791 988 1092 1145 1221 1453 1888

67 160 221 256 399 483 546 644 690 765 978 1126 1193 1237 1517 1889

49 150 214 317 344 435 524 619 698 757 973 1064 1196 1293 1498 1890

14 145 176 315 329 445 487 633 681 739 1044 1057 1136 1225 1501 1891

74 136 183 268 381 451 563 614 728 784 1013 1099 1203 1271 1536 1892

26 105 185 302 332 462 507 612 696 764 1034 1106 1149 1219 1450 1893

36 126 163 320 354 419 561 645 699 796 990 1108 1179 1215 1488 1894

42 82 206 313 338 459 553 599 719 806 1009 1086 1167 1260 1485 1895

53 101 202 275 404 463 535 617 706 732 1018 1129 1180 1233 1512 1896

10 110 226 282 401 478 500 610 652 743 1045 1125 1151 1268 1458 1897

50 137 200 284 362 415 560 572 682 780 1022 1069 1206 1257 1504 1898

54 114 192 262 326 474 512 579 670 740 1027 1123 1140 1282 1438 1899

69 119 174 305 396 444 522 581 683 744 985 1115 1201 1267 1518 1900

6 157 219 301 391 473 528 639 654 759 1008 1097 1199 1239 1520 1901

65 100 199 245 382 409 539 602 709 776 1007 1062 1152 1270 1437 1902

35 99 231 299 351 465 496 598 723 755 1001 1122 1186 1292 1527 1903

64 93 241 291 372 477 536 622 704 805 1040 1074 1204 1276 1444 1904

80 132 167 273 328 443 540 596 702 754 1048 1084 1197 1262 1505 1905

56 140 178 318 347 471 555 588 655 770 1031 1090 1159 1259 1496 1906

68 123 215 298 356 405 492 570 689 746 998 1101 1166 1220 1506 1907

34 90 175 250 383 432 551 615 707 758 1046 1058 1168 1264 1448 1908

62 138 179 260 360 485 521 595 700 804 1049 1098 1146 1254 1530 1909

0 141 194 266 365 440 550 627 662 752 989 1102 1189 1249 1445 1910

23 161 211 277 403 429 566 637 669 729 976 1117 1185 1240 1525 1911

76 148 190 314 346 454 542 643 671 793 1002 1054 1209 1289 1523 1912

31 94 240 274 345 439 554 574 649 766 1032 1113 1183 1230 1486 1913

20 124 189 278 339 411 520 611 692 801 1020 1083 1175 1266 1511 1914

45 112 205 293 378 442 548 571 688 790 1033 1112 1157 1285 1528 1915

30 125 181 310 386 464 486 575 712 735 1004 1128 1202 1294 1521 1916

2 96 193 289 369 448 509 590 686 800 979 1104 1182 1241 1489 1917

33 151 239 259 336 434 562 607 678 772 993 1116 1214 1218 1495 1918

55 85 187 288 384 431 517 586 660 803 974 1082 1144 1223 1497 1919

39 146 162 304 387 472 506 636 705 745 1052 1110 1150 1261 1439 1920

25 144 228 280 327 436 531 585 685 809 1005 1053 1161 1284 1514 1921

22 97 201 292 394 426 516 601 717 795 1039 1071 1198 1283 1510 1922

63 131 236 258 340 421 488 577 727 792 977 1124 1158 1277 1533 1923

27 149 225 286 370 412 519 589 653 753 1050 1079 1162 1236 1508 1924

17 142 170 243 358 461 541 635 664 797 1012 1068 1177 1244 1502 1925

12 104 235 247 371 482 525 583 701 787 1019 1093 1194 1227 1449 1926

3 111 207 300 342 438 511 567 661 782 1021 1078 1173 1274 1526 1927

52 113 238 255 397 457 508 624 665 773 999 1130 1143 1242 1507 1928

73 91 180 244 331 466 549 597 680 742 1042 1081 1172 1245 1538 1929

29 134 164 269 392 413 513 632 697 763 1038 1103 1188 1265 1436 1930

48 130 230 254 390 470 503 621 676 799 982 1087 1164 1252 1446 1931

57 154 227 306 343 475 498 646 726 738 1036 1073 1176 1278 1491 1932

4 128 188 257 377 433 489 631 675 747 1028 1070 1142 1228 1522 1933

61 120 232 279 395 456 538 603 691 774 1010 1111 1170 1216 1452 1934

66 102 222 263 388 455 559 634 667 751 975 1075 1134 1229 1492 1935

24 147 217 249 350 449 515 576 679 756 1035 1065 1211 1280 1503 1936

47 127 208 246 357 408 534 640 725 794 987 1060 1184 1255 1519 1937

46 159 177 287 359 416 543 626 673 737 997 1131 1139 1269 1455 1938

40 89 198 251 337 479 490 623 648 736 1003 1100 1208 1250 1433 1939

79 95 234 321 380 446 547 584 714 730 1014 1121 1153 1248 1500 1940

7 106 173 316 376 414 552 628 687 769 1051 1077 1138 1281 1513 1941

70 143 182 307 400 417 510 618 722 777 1011 1096 1190 1235 1494 1942

37 103 209 276 374 437 533 613 711 760 1015 1105 1141 1253 1456 1943

5 107 186 297 366 424 532 604 656 807 1030 1132 1163 1246 1430 1864

4. based on GRS abnormal LDPC (1944,1621) sign indicating number (4), R=0.8338

1944 324

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

0 81 162 243 324 405 486 567 648 729 810 891 972 1053 1134 1215 1296 1377 1458 15391597 1863

14 102 209 303 338 426 494 598 676 783 886 935 1011 1131 1177 1267 1308 1447 1490 15661687 1865

15 103 210 304 339 427 495 599 677 784 887 936 1012 1132 1178 1268 1309 1448 1491 15671688 1866

16 104 211 305 340 428 496 600 678 785 888 937 1013 1133 1179 1269 1310 1449 1492 15681689 1867

17 105 212 306 341 429 497 601 679 786 889 938 1014 1054 1180 1270 1311 1450 1493 15691690 1868

18 106 213 307 342 430 498 602 680 787 890 939 1015 1055 1181 1271 1312 1451 1494 15701691 1869

19 107 214 308 343 431 499 603 681 788 811 940 1016 1056 1182 1272 1313 1621 1495 15711692 1870

20 108 215 309 344 432 500 604 682 789 812 941 1017 1057 1183 1273 1314 1453 1496 15721693 1871

21 109 216 310 345 433 501 605 683 790 813 942 1018 1058 1184 1274 1315 1622 1497 15731694 1872

22 110 217 311 346 434 502 606 684 791 814 943 1019 1059 1185 1275 1316 1455 1498 15741695 1873

23 111 218 312 347 435 503 607 685 792 815 944 1020 1060 1186 1276 1317 1623 1499 15751696 1874

24 112 219 313 348 436 504 608 686 793 816 945 1021 1061 1187 1277 1318 1624 1500 15761697 1875

25 113 220 314 349 437 505 609 687 794 817 946 1022 1062 1188 1278 1319 1378 1501 15771698 1876

26 114 221 315 350 438 506 610 688 795 818 947 1023 1063 1189 1279 1320 1379 1502 15781699 1877

27 115 222 316 351 439 507 611 689 796 819 948 1024 1064 1190 1280 1321 1380 1503 15791700 1878

28 116 223 317 352 440 508 612 690 797 820 949 1025 1065 1191 1281 1322 1381 1504 15801598 1879

29 117 224 318 353 441 509 613 691 798 821 950 1026 1066 1192 1282 1323 1382 1505 15811600 1880

30 118 225 319 354 442 510 614 692 799 822 951 1027 1067 1193 1283 1324 1383 1506 16271603 1881

31 119 226 320 355 443 511 615 693 800 823 952 1028 1068 1194 1284 1325 1384 1507 15831604 1882

32 120 227 321 356 444 512 616 694 801 824 953 1029 1069 1195 1285 1326 1385 1508 15841606 1883

33 121 228 322 357 445 513 617 695 802 825 954 1030 1070 1196 1286 1327 1386 1509 15851591 1884

34 122 229 323 358 446 514 618 696 803 826 955 1031 1071 1197 1287 1328 1387 1510 15861593 1885

35 123 230 244 359 447 515 619 697 804 827 956 1032 1072 1198 1288 1329 1388 1511 15871596 1886

36 124 231 245 360 448 516 620 698 805 828 957 1033 1073 1199 1289 1330 1389 1512 15881599 1887

37 125 232 246 361 449 517 621 699 806 829 958 1034 1074 1200 1290 1331 1390 1513 15891601 1888

38 126 233 247 362 450 518 622 700 807 830 959 1035 1075 1201 1291 1332 1391 1514 15901602 1889

39 127 234 248 363 451 519 623 701 808 831 960 1036 1076 1202 1292 1333 1392 1515 14521605 1890

40 128 235 249 364 452 520 624 702 809 832 961 1037 1077 1203 1293 1334 1393 1516 16281647 1891

41 129 236 250 365 453 521 625 703 730 833 962 1038 1078 1204 1294 1335 1394 1517 14541607 1892

42 130 237 251 366 454 522 626 704 731 834 963 1039 1079 1205 1295 1336 1395 1518 16291608 1893

43 131 238 252 367 455 523 627 705 732 835 964 1040 1080 1206 1216 1337 1396 1519 16301609 1894

44 132 239 253 368 456 524 628 706 733 836 965 1041 1081 1207 1217 1338 1397 1520 14561648 1895

45 133 240 254 369 457 525 629 707 734 837 966 1042 1082 1208 1218 1339 1398 1521 16311649 1896

46 134 241 255 370 458 526 630 708 735 838 967 1043 1083 1209 1219 1340 1399 1522 16321650 1897

47 135 242 256 371 459 527 631 709 736 839 968 1044 1084 1210 1220 1341 1400 1523 14571611 1898

48 136 163 257 372 460 528 632 710 737 840 969 1045 1085 1211 1221 1342 1401 1524 16331612 1899

49 137 164 258 373 461 529 633 711 738 841 970 1046 1086 1212 1222 1343 1402 1525 15371610 1900

50 138 165 259 374 462 530 634 712 739 842 971 1047 1087 1213 1223 1344 1403 1526 15381613 1901

51 139 166 260 375 463 531 635 713 740 843 892 1048 1088 1214 1224 1345 1404 1527 16341651 1902

52 140 167 261 376 464 532 636 714 741 844 893 1049 1089 1135 1225 1346 1405 1528 16351652 1903

53 141 168 262 377 465 533 637 715 742 845 894 1050 1090 1136 1226 1347 1406 1529 15821653 1904

54 142 169 263 378 466 534 638 716 743 846 895 1051 1091 1137 1227 1348 1407 1530 16361654 1905

55 143 170 264 379 467 535 639 717 744 847 896 1052 1092 1138 1228 1349 1408 1531 15921655 1906

56 144 171 265 380 468 536 640 718 745 848 897 973 1093 1139 1229 1350 1409 1532 16371656 1907

57 145 172 266 381 469 537 641 719 746 849 898 974 1094 1140 1230 1351 1410 1533 15941614 1908

58 146 173 267 382 470 538 642 720 747 850 899 975 1095 1141 1231 1352 1411 1534 16381657 1909

59 147 174 268 383 471 539 643 721 748 851 900 976 1096 1142 1232 1353 1412 1535 16391658 1910

60 148 175 269 384 472 540 644 722 749 852 901 977 1097 1143 1233 1354 1413 1536 15951659 1911

61 149 176 270 385 473 541 645 723 750 853 902 978 1098 1144 1234 1355 1414 1625 16401660 1912

62 150 177 271 386 474 542 646 724 751 854 903 979 1099 1145 1235 1356 1415 1626 16411661 1913

63 151 178 272 387 475 543 647 725 752 855 904 980 1100 1146 1236 1357 1416 1459 16421615 1914

64 152 179 273 388 476 544 568 726 753 856 905 981 1101 1147 1237 1358 1417 1460 16431662 1915

65 153 180 274 389 477 545 569 727 754 857 906 982 1102 1148 1238 1359 1418 1461 16441616 1916

66 154 181 275 390 478 546 570 728 755 858 907 983 1103 1149 1239 1360 1419 1462 16451663 1917

67 155 182 276 391 479 547 571 649 756 859 908 984 1104 1150 1240 1361 1420 1463 16461664 1918

68 156 183 277 392 480 548 572 650 757 860 909 985 1105 1151 1241 1362 1421 1464 15401617 1919

69 157 184 278 393 481 549 573 651 758 861 910 986 1106 1152 1242 1363 1422 1465 15411665 1920

70 158 185 279 394 482 550 574 652 759 862 911 987 1107 1153 1243 1364 1423 1466 15421666 1921

71 159 186 280 395 483 551 575 653 760 863 912 988 1108 1154 1244 1365 1424 1467 15431667 1922

72 160 187 281 396 484 552 576 654 761 864 913 989 1109 1155 1245 1366 1425 1468 15441618 1923

73 161 188 282 397 485 553 577 655 762 865 914 990 1110 1156 1246 1367 1426 1469 15451619 1924

74 82 189 283 398 406 554 578 656 763 866 915 991 1111 1157 1247 1368 1427 1470 15461668 1925

75 83 190 284 399 407 555 579 657 764 867 916 992 1112 1158 1248 1369 1428 1471 15471669 1926

76 84 191 285 400 408 556 580 658 765 868 917 993 1113 1159 1249 1370 1429 1472 15481670 1927

77 85 192 286 401 409 557 581 659 766 869 918 994 1114 1160 1250 1371 1430 1473 15491671 1928

78 86 193 287 402 410 558 582 660 767 870 919 995 1115 1161 1251 1372 1431 1474 15501672 1929

79 87 194 288 403 411 559 583 661 768 871 920 996 1116 1162 1252 1373 1432 1475 15511673 1930

80 88 195 289 404 412 560 584 662 769 872 921 997 1117 1163 1253 1374 1433 1476 15521674 1931

1 89 196 290 325 413 561 585 663 770 873 922 998 1118 1164 1254 1375 1434 1477 15531675 1932

2 90 197 291 326 414 562 586 664 771 874 923 999 1119 1165 1255 1376 1435 1478 15541676 1933

3 91 198 292 327 415 563 587 665 772 875 924 1000 1120 1166 1256 1297 1436 1479 15551677 1934

4 92 199 293 328 416 564 588 666 773 876 925 1001 1121 1167 1257 1298 1437 1480 15561678 1935

5 93 200 294 329 417 565 589 667 774 877 926 1002 1122 1168 1258 1299 1438 1481 15571679 1936

6 94 201 295 330 418 566 590 668 775 878 927 1003 1123 1169 1259 1300 1439 1482 15581680 1937

7 95 202 296 331 419 487 591 669 776 879 928 1004 1124 1170 1260 1301 1440 1483 15591620 1938

8 96 203 297 332 420 488 592 670 777 880 929 1005 1125 1171 1261 1302 1441 1484 15601681 1939

9 97 204 298 333 421 489 593 671 778 881 930 1006 1126 1172 1262 1303 1442 1485 15611682 1940

10 98 205 299 334 422 490 594 672 779 882 931 1007 1127 1173 1263 1304 1443 1486 15621683 1941

11 99 206 300 335 423 491 595 673 780 883 932 1008 1128 1174 1264 1305 1444 1487 15631684 1942

12 100 207 301 336 424 492 596 674 781 884 933 1009 1129 1175 1265 1306 1445 1488 15641685 1943

13 101 208 302 337 425 493 597 675 782 885 934 1010 1130 1176 1266 1307 1446 1489 15651686 1864

15 148 220 319 394 424 501 616 706 785 837 943 976 1063 1203 1252 1356 1418 1527 14521602 1778

67 113 241 281 331 448 519 574 726 807 873 914 1037 1090 1211 1234 1322 1453 1459 15891604 1727

55 129 200 286 337 423 487 601 712 757 857 961 1013 1084 1201 1230 1372 1388 1528 15841695 1736

68 105 208 244 348 437 490 578 709 745 843 929 1025 1123 1196 1254 1321 1405 1473 15751667 1763

39 117 191 267 385 418 510 583 670 758 840 932 991 1131 1187 1228 1337 1384 1626 15441685 1740

14 83 238 266 345 416 497 621 714 809 881 952 1019 1114 1156 1220 1313 1434 1510 15651612 1745

28 111 206 260 349 449 523 644 704 784 845 939 972 1081 1177 1282 1325 1383 1461 15371664 1703

9 81 209 299 364 483 553 643 699 798 835 965 980 1129 1213 1247 1371 1399 1483 15401670 1726

7 152 229 307 381 421 541 637 690 779 830 915 1033 1132 1152 1227 1319 1455 1467 15491696 1725

40 125 216 290 360 414 554 596 659 777 821 903 988 1072 1161 1259 1296 1387 1533 15761674 1719

74 160 242 257 330 456 525 604 680 730 870 916 977 1059 1188 1269 1360 1433 1530 15531679 1758

12 149 192 305 359 463 500 587 716 764 811 967 1002 1085 1165 1275 1333 1381 1491 15581609 1766

5 94 180 308 375 465 514 634 655 782 847 942 987 1115 1170 1286 1368 1377 1535 14561663 1749

47 159 193 248 351 443 495 602 664 775 866 956 1039 1103 1208 1243 1357 1422 1525 16461616 1716

54 131 164 315 363 406 493 605 691 737 875 937 990 1116 1151 1283 1302 1395 1520 16451658 1764

56 82 219 261 329 482 526 625 668 744 822 935 1012 1087 1150 1287 1367 1430 1511 15951691 1767

34 104 233 291 357 426 560 612 673 746 879 968 996 1062 1144 1222 1339 1419 1480 15711699 1707

77 88 214 279 324 480 498 638 711 804 884 922 982 1076 1183 1239 1370 1444 1501 15791682 1774

73 154 212 292 398 472 491 588 654 767 842 940 979 1057 1191 1218 1312 1429 1625 15621656 1720

17 151 165 263 371 454 533 576 653 763 865 933 1020 1055 1174 1268 1376 1401 1476 15941697 1750

71 112 199 318 326 419 540 589 727 787 864 895 984 1088 1141 1217 1362 1432 1485 15771700 1738

63 156 217 252 395 479 542 640 686 761 858 902 974 1122 1189 1233 1359 1622 1512 15801611 1751

45 146 210 313 340 431 520 615 694 753 817 904 1049 1060 1192 1289 1320 1438 1489 16331593 1722

10 141 172 311 325 441 563 629 677 735 825 962 1040 1133 1212 1221 1364 1424 1494 15871659 1777

70 132 179 264 377 447 559 610 724 780 888 925 1009 1095 1199 1267 1354 1421 1532 16401683 1711

22 101 181 298 328 458 503 608 692 760 855 921 1030 1102 1145 1295 1349 1382 1475 15631672 1772

32 122 239 316 350 415 557 641 695 792 823 945 986 1104 1175 1215 1340 1426 1474 15511684 1770

38 158 202 309 334 455 549 595 715 802 826 919 1005 1082 1163 1256 1309 1416 1468 15641661 1723

49 97 198 271 400 459 531 613 702 808 846 911 1014 1125 1176 1229 1330 1411 1507 16421601 1757

6 106 222 278 397 474 496 606 728 739 833 893 1041 1121 1147 1264 1366 1402 1515 15901651 1775

46 133 196 280 358 411 556 568 678 776 859 938 1018 1065 1202 1253 1305 1451 1498 16351606 1768

50 110 188 258 402 470 508 575 666 736 889 918 1023 1119 1136 1278 1314 1392 1465 15851603 1730

65 115 170 301 392 440 518 577 679 740 877 950 981 1111 1197 1263 1341 1428 1513 15831615 1737

2 153 215 297 387 469 524 635 650 755 890 960 1004 1093 1195 1235 1318 1447 1516 16431690 1739

61 96 195 321 378 485 535 598 705 772 861 966 1003 1058 1148 1266 1323 1623 1536 15701596 1717

31 95 227 295 347 461 492 594 719 751 836 897 997 1118 1182 1288 1361 1403 1523 15881598 1760

60 89 237 287 368 473 532 618 700 801 850 934 1036 1070 1200 1272 1304 1380 1469 15811666 1756

76 128 163 269 404 439 536 592 698 750 831 894 1044 1080 1193 1258 1303 1385 1499 15431671 1780

52 136 174 314 343 467 551 584 651 766 829 898 1027 1086 1155 1255 1297 1423 1487 15501673 1754

64 119 211 294 352 405 488 646 685 742 862 913 994 1097 1162 1216 1336 1446 1500 15521614 1746

30 86 171 246 379 428 547 611 703 754 816 930 1042 1054 1164 1260 1344 1445 1471 16381693 1728

58 134 175 256 356 481 517 591 696 800 834 909 1045 1094 1142 1250 1327 1439 1526 15731689 1773

0 137 190 262 361 436 546 623 658 748 827 959 985 1098 1185 1245 1374 1398 1460 15691647 1753

19 157 207 273 399 425 562 633 665 729 869 908 1052 1113 1181 1236 1342 1406 1521 14541687 1705

72 144 186 310 342 450 538 639 667 789 876 924 998 1130 1205 1285 1345 1389 1519 15671680 1715

27 90 236 270 341 435 550 570 725 762 878 900 1028 1109 1179 1226 1365 1436 1472 15591617 1721

16 120 185 274 335 407 516 607 688 797 856 912 1016 1079 1171 1262 1352 1404 1506 15411591 1732

41 108 201 289 374 438 544 647 684 786 819 958 1029 1108 1153 1281 1298 1407 1524 15861686 1769

26 121 177 306 382 460 486 571 708 731 815 906 1000 1124 1198 1290 1328 1427 1517 15661649 1729

78 92 189 285 365 444 505 586 682 796 839 891 975 1100 1178 1237 1316 1414 1479 16321655 1733

29 147 235 255 332 430 558 603 674 768 813 947 989 1112 1210 1294 1329 1440 1486 16371660 1748

51 161 183 284 380 427 513 582 656 799 885 920 1050 1078 1140 1219 1300 1390 1488 16411618 1765

35 142 162 300 383 468 502 632 701 741 867 955 1048 1106 1146 1257 1355 1378 1466 15451600 1744

21 140 224 276 403 432 527 581 681 805 832 944 1001 1053 1157 1280 1369 1391 1509 16271665 1714

18 93 197 288 390 422 512 597 713 791 812 969 1035 1067 1194 1279 1350 1442 1505 15421619 1743

59 127 232 254 336 417 564 573 723 788 844 954 973 1120 1154 1273 1348 1417 1529 15461681 1759

23 145 221 282 366 408 515 585 649 749 854 926 1046 1075 1158 1232 1301 1431 1503 15611698 1735

13 138 166 243 354 457 537 631 660 793 860 957 1008 1064 1173 1240 1335 1412 1495 15781678 1747

8 100 231 323 367 478 521 579 697 783 871 899 1015 1089 1190 1223 1353 1410 1477 15571654 1713

79 107 203 296 338 434 507 567 657 778 828 963 1017 1074 1169 1270 1346 1443 1522 15921677 1741

48 109 234 251 393 453 504 620 661 769 868 949 995 1126 1139 1238 1308 1397 1502 15561692 1701

69 87 176 320 327 462 545 593 676 738 872 946 1038 1077 1168 1241 1315 1415 1534 15721669 1702

25 130 240 265 388 409 509 628 693 759 887 907 1034 1099 1184 1261 1317 1408 1464 15481620 1755

44 126 226 250 386 466 499 617 672 795 824 951 978 1083 1160 1248 1375 1450 1470 15601653 1710

53 150 223 302 339 471 494 642 722 734 883 941 1032 1069 1172 1274 1338 1624 1481 16361675 1779

80 124 184 253 373 429 565 627 671 743 853 936 1024 1066 1138 1224 1334 1379 1518 15541597 1724

57 116 228 275 391 452 534 599 687 770 882 927 1006 1107 1166 1292 1358 1437 1478 15391608 1709

62 98 218 259 384 451 555 630 663 747 818 896 1051 1071 1134 1225 1332 1400 1482 16301688 1761

20 143 213 245 346 445 511 572 675 752 874 917 1031 1061 1207 1276 1324 1396 1497 15681613 1712

43 123 204 322 353 484 530 636 721 790 886 953 983 1056 1180 1251 1306 1420 1514 16341605 1734

42 155 173 283 355 412 539 622 669 733 852 892 993 1127 1135 1265 1351 1394 1493 16281662 1718

36 85 194 247 333 475 566 619 648 732 880 901 999 1096 1204 1246 1331 1386 1463 16441610 1704

75 91 230 317 376 442 543 580 710 806 810 928 1010 1117 1149 1244 1363 1448 1492 15381694 1781

3 102 169 312 372 410 548 624 683 765 841 905 1047 1073 1214 1277 1373 1413 1508 15741652 1742

66 139 178 303 396 413 506 614 718 773 814 910 1007 1092 1186 1231 1299 1393 1484 15821668 1706

33 99 205 272 370 433 529 609 707 756 849 948 1011 1101 1137 1249 1310 1425 1496 15471657 1776

1 103 182 293 362 420 528 600 652 803 838 971 1026 1128 1159 1242 1347 1435 1462 16391648 1771

4 118 187 249 344 446 522 569 717 771 863 970 1043 1105 1143 1284 1307 1441 1490 16311607 1762

24 135 225 268 389 476 561 590 689 774 848 964 1022 1110 1209 1291 1311 1621 1458 16291676 1731

11 114 168 277 369 464 489 626 720 794 820 923 992 1068 1206 1293 1326 1409 1531 15551650 1752

37 84 167 304 401 477 552 645 662 781 851 931 1021 1091 1167 1271 1343 1449 1504 14571599 1708

16 149 221 320 395 425 502 617 707 786 838 944 977 1064 1204 1253 1357 1419 1528 16281779 1785

38 85 168 305 402 478 553 646 663 782 852 932 1022 1092 1168 1272 1344 1450 1505 16331709 1810

68 114 242 282 332 449 520 575 727 808 874 915 1038 1091 1212 1235 1323 1622 1460 15901728 1844

56 130 201 287 338 424 488 602 713 758 858 962 1014 1085 1202 1231 1373 1389 1529 15851737 1862

69 106 209 245 349 438 491 579 710 746 844 930 1026 1124 1197 1255 1322 1406 1474 15761764 1855

40 118 192 268 386 419 511 584 671 759 841 933 992 1132 1188 1229 1338 1385 1459 1545 1741 1817

15 84 239 267 346 417 498 622 715 730 882 953 1020 1115 1157 1221 1314 1435 1511 15661746 1824

29 112 207 261 350 450 524 645 705 785 846 940 972 1082 1178 1283 1326 1384 1462 15381704 1826

10 81 210 300 365 484 554 644 700 799 836 966 981 1130 1214 1248 1372 1400 1484 15411727 1804

8 153 230 308 382 422 542 638 691 780 831 916 1034 1133 1153 1228 1320 1623 1468 15501726 1847

41 126 217 291 361 415 555 597 660 778 822 904 989 1073 1162 1260 1296 1388 1534 15771720 1843

75 161 163 258 331 457 526 605 681 731 871 917 978 1060 1189 1270 1361 1434 1531 15541759 1787

13 150 193 306 360 464 501 588 717 765 812 968 1003 1086 1166 1276 1334 1382 1492 15591767 1841

6 95 181 309 376 466 515 635 656 783 848 943 988 1116 1171 1287 1369 1377 1536 16311750 1833

48 160 194 249 352 444 496 603 665 776 867 957 1040 1104 1209 1244 1358 1423 1526 15401717 1815

55 132 165 316 364 407 494 606 692 738 876 938 991 1117 1152 1284 1303 1396 1521 16461765 1860

57 83 220 262 330 483 527 626 669 745 823 936 1013 1088 1151 1288 1368 1431 1512 16401768 1840

35 105 234 292 358 427 561 613 674 747 880 969 997 1063 1145 1223 1340 1420 1481 15721708 1792

78 89 215 280 324 481 499 639 712 805 885 923 983 1077 1184 1240 1371 1445 1502 15801775 1802

74 155 213 293 399 473 492 589 655 768 843 941 980 1058 1192 1219 1313 1430 1626 15631721 1808

18 152 166 264 372 455 534 577 654 764 866 934 1021 1056 1175 1269 1297 1402 1477 16381751 1819

72 113 200 319 327 420 541 590 728 788 865 896 985 1089 1142 1218 1363 1433 1486 15781739 1856

64 157 218 253 396 480 543 641 687 762 859 903 975 1123 1190 1234 1360 1455 1513 15811752 1816

46 147 211 314 341 432 521 616 695 754 818 905 1050 1061 1193 1290 1321 1439 1490 15371723 1820

11 142 173 312 326 442 564 630 678 736 826 963 1041 1054 1213 1222 1365 1425 1495 15881778 1835

71 133 180 265 378 448 560 611 725 781 889 926 1010 1096 1200 1268 1355 1422 1533 16411712 1852

23 102 182 299 329 459 504 609 693 761 856 922 1031 1103 1146 1216 1350 1383 1476 15641773 1831

33 123 240 317 351 416 558 642 696 793 824 946 987 1105 1176 1215 1341 1427 1475 15521771 1801

39 159 203 310 335 456 550 596 716 803 827 920 1006 1083 1164 1257 1310 1417 1469 15651724 1830

50 98 199 272 401 460 532 614 703 809 847 912 1015 1126 1177 1230 1331 1412 1508 16431758 1846

7 107 223 279 398 475 497 607 649 740 834 894 1042 1122 1148 1265 1367 1403 1516 14521776 1822

47 134 197 281 359 412 557 569 679 777 860 939 1019 1066 1203 1254 1306 1621 1499 15821769 1834

51 111 189 259 403 471 509 576 667 737 890 919 1024 1120 1137 1279 1315 1393 1466 15861731 1800

66 116 171 302 393 441 519 578 680 741 878 951 982 1112 1198 1264 1342 1429 1514 15841738 1828

3 154 216 298 388 470 525 636 651 756 811 961 1005 1094 1196 1236 1319 1448 1517 16441740 1782

62 97 196 322 379 406 536 599 706 773 862 967 1004 1059 1149 1267 1324 1624 1625 15711718 1789

32 96 228 296 348 462 493 595 720 752 837 898 998 1119 1183 1289 1362 1404 1524 15891761 1842

61 90 238 288 369 474 533 619 701 802 851 935 1037 1071 1201 1273 1305 1381 1470 16271757 1797

77 129 164 270 325 440 537 593 699 751 832 895 1045 1081 1194 1259 1304 1386 1500 15441781 1786

53 137 175 315 344 468 552 585 652 767 830 899 1028 1087 1156 1256 1298 1424 1488 15511755 1811

65 120 212 295 353 405 489 647 686 743 863 914 995 1098 1163 1217 1337 1447 1501 15531747 1796

31 87 172 247 380 429 548 612 704 755 817 931 1043 1055 1165 1261 1345 1446 1472 16391729 1848

59 135 176 257 357 482 518 592 697 801 835 910 1046 1095 1143 1251 1328 1440 1527 15741774 1799

0 138 191 263 362 437 547 624 659 749 828 960 986 1099 1186 1246 1375 1399 1461 15701754 1821

20 158 208 274 400 426 563 634 666 729 870 909 973 1114 1182 1237 1343 1407 1522 16291706 1805

73 145 187 311 343 451 539 640 668 790 877 925 999 1131 1206 1286 1346 1390 1520 15681716 1791

28 91 237 271 342 436 551 571 726 763 879 901 1029 1110 1180 1227 1366 1437 1473 15601722 1788

17 121 186 275 336 408 517 608 689 798 857 913 1017 1080 1172 1263 1353 1405 1507 15421733 1829

42 109 202 290 375 439 545 568 685 787 820 959 1030 1109 1154 1282 1299 1408 1525 15871770 1793

27 122 178 307 383 461 486 572 709 732 816 907 1001 1125 1199 1291 1329 1428 1518 15671730 1783

79 93 190 286 366 445 506 587 683 797 840 891 976 1101 1179 1238 1317 1415 1480 14571734 1858

30 148 236 256 333 431 559 604 675 769 814 948 990 1113 1211 1295 1330 1441 1487 15941749 1849

52 82 184 285 381 428 514 583 657 800 886 921 1051 1079 1141 1220 1301 1391 1489 16421766 1818

36 143 162 301 384 469 503 633 702 742 868 956 1049 1107 1147 1258 1356 1379 1467 15461745 1839

22 141 225 277 404 433 528 582 682 806 833 945 1002 1053 1158 1281 1370 1392 1510 15831715 1795

19 94 198 289 391 423 513 598 714 792 813 970 1036 1068 1195 1280 1351 1443 1506 15431744 1814

60 128 233 255 337 418 565 574 724 789 845 955 974 1121 1155 1274 1349 1418 1530 15471760 1823

24 146 222 283 367 409 516 586 650 750 855 927 1047 1076 1159 1233 1302 1432 1504 15621736 1850

14 139 167 243 355 458 538 632 661 794 861 958 1009 1065 1174 1241 1336 1413 1496 15791748 1827

9 101 232 244 368 479 522 580 698 784 872 900 1016 1090 1191 1224 1354 1411 1478 15581714 1832

80 108 204 297 339 435 508 567 658 779 829 964 1018 1075 1170 1271 1347 1444 1523 16371742 1790

49 110 235 252 394 454 505 621 662 770 869 950 996 1127 1140 1239 1309 1398 1503 15571701 1813

70 88 177 321 328 463 546 594 677 739 873 947 1039 1078 1169 1242 1316 1416 1535 15731703 1812

26 131 241 266 389 410 510 629 694 760 888 908 1035 1100 1185 1262 1318 1409 1465 15491756 1806

45 127 227 251 387 467 500 618 673 796 825 952 979 1084 1161 1249 1376 1451 1471 15611711 1845

54 151 224 303 340 472 495 643 723 735 884 942 1033 1070 1173 1275 1339 1378 1482 15921780 1853

1 125 185 254 374 430 566 628 672 744 854 937 1025 1067 1139 1225 1335 1380 1519 15551725 1836

58 117 229 276 392 453 535 600 688 771 883 928 1007 1108 1167 1293 1359 1438 1479 15391710 1803

63 99 219 260 385 452 556 631 664 748 819 897 1052 1072 1134 1226 1333 1401 1483 14561762 1851

21 144 214 246 347 446 512 573 676 753 875 918 1032 1062 1208 1277 1325 1397 1498 15691713 1854

44 124 205 323 354 485 531 637 722 791 887 954 984 1057 1181 1252 1307 1421 1515 16351735 1794

43 156 174 284 356 413 540 623 670 734 853 893 994 1128 1136 1266 1352 1395 1494 14541719 1861

37 86 195 248 334 476 487 620 648 733 881 902 1000 1097 1205 1247 1332 1387 1464 16451705 1807

76 92 231 318 377 443 544 581 711 807 810 929 1011 1118 1150 1245 1364 1449 1493 16341702 1837

4 103 170 313 373 411 549 625 684 766 842 906 1048 1074 1135 1278 1374 1414 1509 15751743 1825

67 140 179 304 397 414 507 615 719 774 815 911 1008 1093 1187 1232 1300 1394 1485 16361707 1838

34 100 206 273 371 434 530 610 708 757 850 949 1012 1102 1138 1250 1311 1426 1497 15481777 1809

2 104 183 294 363 421 529 601 653 804 839 892 1027 1129 1160 1243 1348 1436 1463 15951772 1784

5 119 188 250 345 447 523 570 718 772 864 971 1044 1106 1144 1285 1308 1442 1491 16321763 1798

25 136 226 269 390 477 562 591 690 775 849 965 1023 1111 1210 1292 1312 1453 1458 16301732 1859

12 115 169 278 370 465 490 627 721 795 821 924 993 1069 1207 1294 1327 1410 1532 15561753 1857

17 150 222 321 396 426 503 618 708 787 839 945 978 1065 1205 1254 1358 1420 1529 14541786 1863

13 116 170 279 371 466 491 628 722 796 822 925 994 1070 1208 1295 1328 1411 1533 15571858 1865

39 86 169 306 403 479 554 647 664 783 853 933 1023 1093 1169 1273 1345 1451 1506 15371811 1866

69 115 163 283 333 450 521 576 728 809 875 916 1039 1092 1213 1236 1324 1455 1461 14521845 1867

57 131 202 288 339 425 489 603 714 759 859 963 1015 1086 1203 1232 1374 1390 1530 15861783 1868

70 107 210 246 350 439 492 580 711 747 845 931 1027 1125 1198 1256 1323 1407 1475 15771856 1869

41 119 193 269 387 420 512 585 672 760 842 934 993 1133 1189 1230 1339 1386 1460 15461818 1870

16 85 240 268 347 418 499 623 716 731 883 954 1021 1116 1158 1222 1315 1436 1512 15671825 1871

30 113 208 262 351 451 525 646 706 786 847 941 972 1083 1179 1284 1327 1385 1463 16341827 1872

11 81 211 301 366 485 555 645 701 800 837 967 982 1131 1135 1249 1373 1401 1485 15421805 1873

9 154 231 309 383 423 543 639 692 781 832 917 1035 1054 1154 1229 1321 1624 1469 15511848 1874

42 127 218 292 362 416 556 598 661 779 823 905 990 1074 1163 1261 1296 1389 1535 15781844 1875

76 82 164 259 332 458 527 606 682 732 872 918 979 1061 1190 1271 1362 1435 1532 15551788 1876

14 151 194 307 361 465 502 589 718 766 813 969 1004 1087 1167 1277 1335 1383 1493 15601842 1877

7 96 182 310 377 467 516 636 657 784 849 944 989 1117 1172 1288 1370 1377 1625 16321834 1878

49 161 195 250 353 445 497 604 666 777 868 958 1041 1105 1210 1245 1359 1424 1527 15411816 1879

56 133 166 317 365 408 495 607 693 739 877 939 992 1118 1153 1285 1304 1397 1522 15401861 1880

58 84 221 263 331 484 528 627 670 746 824 937 1014 1089 1152 1289 1369 1432 1513 16411841 1881

36 106 235 293 359 428 562 614 675 748 881 970 998 1064 1146 1224 1341 1421 1482 15731793 1882

79 90 216 281 324 482 500 640 713 806 886 924 984 1078 1185 1241 1372 1446 1503 15811803 1883

75 156 214 294 400 474 493 590 656 769 844 942 981 1059 1193 1220 1314 1431 1459 15641809 1884

19 153 167 265 373 456 535 578 655 765 867 935 1022 1057 1176 1270 1298 1403 1478 16391820 1885

73 114 201 320 328 421 542 591 649 789 866 897 986 1090 1143 1219 1364 1434 1487 15791857 1886

65 158 219 254 397 481 544 642 688 763 860 904 976 1124 1191 1235 1361 1623 1514 16271817 1887

47 148 212 315 342 433 522 617 696 755 819 906 1051 1062 1194 1291 1322 1440 1491 15381821 1888

12 143 174 313 327 443 565 631 679 737 827 964 1042 1055 1214 1223 1366 1426 1496 15891836 1889

72 134 181 266 379 449 561 612 726 782 890 927 1011 1097 1201 1269 1356 1423 1534 16421853 1890

24 103 183 300 330 460 505 610 694 762 857 923 1032 1104 1147 1217 1351 1384 1477 15651832 1891

34 124 241 318 352 417 559 643 697 794 825 947 988 1106 1177 1215 1342 1428 1476 15531802 1892

40 160 204 311 336 457 551 597 717 804 828 921 1007 1084 1165 1258 1311 1418 1470 15661831 1893

51 99 200 273 402 461 533 615 704 730 848 913 1016 1127 1178 1231 1332 1413 1509 16441847 1894

8 108 224 280 399 476 498 608 650 741 835 895 1043 1123 1149 1266 1368 1404 1517 16281823 1895

48 135 198 282 360 413 558 570 680 778 861 940 1020 1067 1204 1255 1307 1453 1500 16361835 1896

52 112 190 260 404 472 510 577 668 738 811 920 1025 1121 1138 1280 1316 1394 1467 15871801 1897

67 117 172 303 394 442 520 579 681 742 879 952 983 1113 1199 1265 1343 1430 1515 15851829 1898

4 155 217 299 389 471 526 637 652 757 812 962 1006 1095 1197 1237 1320 1449 1518 16451782 1899

63 98 197 323 380 407 537 600 707 774 863 968 1005 1060 1150 1268 1325 1378 1626 15721790 1900

33 97 229 297 349 463 494 596 721 753 838 899 999 1120 1184 1290 1363 1405 1525 15901843 1901

62 91 239 289 370 475 534 620 702 803 852 936 1038 1072 1202 1274 1306 1382 1471 15831798 1902

78 130 165 271 326 441 538 594 700 752 833 896 1046 1082 1195 1260 1305 1387 1501 15451787 1903

54 138 176 316 345 469 553 586 653 768 831 900 1029 1088 1157 1257 1299 1425 1489 15521812 1904

66 121 213 296 354 405 490 568 687 744 864 915 996 1099 1164 1218 1338 1448 1502 15541797 1905

32 88 173 248 381 430 549 613 705 756 818 932 1044 1056 1166 1262 1346 1447 1473 15951849 1906

60 136 177 258 358 483 519 593 698 802 836 911 1047 1096 1144 1252 1329 1441 1528 15751800 1907

0 139 192 264 363 438 548 625 660 750 829 961 987 1100 1187 1247 1376 1400 1462 15711822 1908

21 159 209 275 401 427 564 635 667 729 871 910 974 1115 1183 1238 1344 1408 1523 16301806 1909

74 146 188 312 344 452 540 641 669 791 878 926 1000 1132 1207 1287 1347 1391 1521 15691792 1910

29 92 238 272 343 437 552 572 727 764 880 902 1030 1111 1181 1228 1367 1438 1474 15611789 1911

18 122 187 276 337 409 518 609 690 799 858 914 1018 1081 1173 1264 1354 1406 1508 15431830 1912

43 110 203 291 376 440 546 569 686 788 821 960 1031 1110 1155 1283 1300 1409 1526 15881794 1913

28 123 179 308 384 462 486 573 710 733 817 908 1002 1126 1200 1292 1330 1429 1519 15681784 1914

80 94 191 287 367 446 507 588 684 798 841 891 977 1102 1180 1239 1318 1416 1481 16331859 1915

31 149 237 257 334 432 560 605 676 770 815 949 991 1114 1212 1216 1331 1442 1488 16381850 1916

53 83 185 286 382 429 515 584 658 801 887 922 1052 1080 1142 1221 1302 1392 1490 16431819 1917

37 144 162 302 385 470 504 634 703 743 869 957 1050 1108 1148 1259 1357 1380 1468 15471840 1918

23 142 226 278 325 434 529 583 683 807 834 946 1003 1053 1159 1282 1371 1393 1511 15841796 1919

20 95 199 290 392 424 514 599 715 793 814 971 1037 1069 1196 1281 1352 1444 1507 15441815 1920

61 129 234 256 338 419 566 575 725 790 846 956 975 1122 1156 1275 1350 1419 1531 15481824 1921

25 147 223 284 368 410 517 587 651 751 856 928 1048 1077 1160 1234 1303 1433 1505 15631851 1922

15 140 168 243 356 459 539 633 662 795 862 959 1010 1066 1175 1242 1337 1414 1497 15801828 1923

10 102 233 245 369 480 523 581 699 785 873 901 1017 1091 1192 1225 1355 1412 1479 15591833 1924

1 109 205 298 340 436 509 567 659 780 830 965 1019 1076 1171 1272 1348 1445 1524 15941791 1925

50 111 236 253 395 455 506 622 663 771 870 951 997 1128 1141 1240 1310 1399 1504 15581814 1926

71 89 178 322 329 464 547 595 678 740 874 948 1040 1079 1170 1243 1317 1417 1536 15741813 1927

27 132 242 267 390 411 511 630 695 761 889 909 1036 1101 1186 1263 1319 1410 1466 15501807 1928

46 128 228 252 388 468 501 619 674 797 826 953 980 1085 1162 1250 1297 1621 1472 15621846 1929

55 152 225 304 341 473 496 644 724 736 885 943 1034 1071 1174 1276 1340 1379 1483 16371854 1930

2 126 186 255 375 431 487 629 673 745 855 938 1026 1068 1140 1226 1336 1381 1520 15561837 1931

59 118 230 277 393 454 536 601 689 772 884 929 1008 1109 1168 1294 1360 1439 1480 15391804 1932

64 100 220 261 386 453 557 632 665 749 820 898 973 1073 1134 1227 1334 1402 1484 16311852 1933

22 145 215 247 348 447 513 574 677 754 876 919 1033 1063 1209 1278 1326 1398 1499 15701855 1934

45 125 206 244 355 406 532 638 723 792 888 955 985 1058 1182 1253 1308 1422 1516 15821795 1935

44 157 175 285 357 414 541 624 671 735 854 894 995 1129 1137 1267 1353 1396 1495 16291862 1936

38 87 196 249 335 477 488 621 648 734 882 903 1001 1098 1206 1248 1333 1388 1465 16461808 1937

77 93 232 319 378 444 545 582 712 808 810 930 1012 1119 1151 1246 1365 1450 1494 16351838 1938

5 104 171 314 374 412 550 626 685 767 843 907 1049 1075 1136 1279 1375 1415 1510 15761826 1939

68 141 180 305 398 415 508 616 720 775 816 912 1009 1094 1188 1233 1301 1395 1486 15921839 1940

35 101 207 274 372 435 531 611 709 758 851 950 1013 1103 1139 1251 1312 1427 1498 15491810 1941

3 105 184 295 364 422 530 602 654 805 840 893 1028 1130 1161 1244 1349 1437 1464 16401785 1942

6 120 189 251 346 448 524 571 719 773 865 892 1045 1107 1145 1286 1309 1443 1492 14571799 1943

26 137 227 270 391 478 563 592 691 776 850 966 1024 1112 1211 1293 1313 1622 1458 14561860 1864

5. based on GRS abnormal LDPC (1944,1296) sign indicating number (5), R=2/3

1944 648

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

12 12 12 12 12 12 12 12 12 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12

0 81 162 243 324 486 648 810 972 1134 1296 1377

14 102 209 303 338 494 676 886 1011 1177 1308 1447

15 103 210 304 339 495 677 887 1012 1178 1309 1448

16 104 211 305 340 496 678 888 1013 1179 1310 1449

17 105 212 306 341 497 679 889 1014 1180 1311 1450

18 106 213 307 342 498 680 890 1015 1181 1312 1451

19 107 214 308 343 499 681 811 1016 1182 1313 1452

20 108 215 309 344 500 682 812 1017 1183 1314 1453

21 109 216 310 345 501 683 813 1018 1184 1315 1454

22 110 217 311 346 502 684 814 1019 1185 1316 1455

23 111 218 312 347 503 685 815 1020 1186 1317 1456

24 112 219 313 348 504 686 816 1021 1187 1318 1457

25 113 220 314 349 505 687 817 1022 1188 1319 1378

26 114 221 315 350 506 688 818 1023 1189 1320 1379

27 115 222 316 351 507 689 819 1024 1190 1321 1380

28 116 223 317 352 508 690 820 1025 1191 1322 1381

29 117 224 318 353 509 691 821 1026 1192 1323 1382

30 118 225 319 354 510 692 822 1027 1193 1324 1383

31 119 226 320 355 511 693 823 1028 1194 1325 1384

32 120 227 321 356 512 694 824 1029 1195 1326 1385

33 121 228 322 357 513 695 825 1030 1196 1327 1386

34 122 229 323 358 514 696 826 1031 1197 1328 1387

35 123 230 244 359 515 697 827 1032 1198 1329 1388

36 124 231 245 360 516 698 828 1033 1199 1330 1389

37 125 232 246 361 517 699 829 1034 1200 1331 1390

38 126 233 247 362 518 700 830 1035 1201 1332 1391

39 127 234 248 363 519 701 831 1036 1202 1333 1392

40 128 235 249 364 520 702 832 1037 1203 1334 1393

41 129 236 250 365 521 703 833 1038 1204 1335 1394

42 130 237 251 366 522 704 834 1039 1205 1336 1395

43 131 238 252 367 523 705 835 1040 1206 1337 1396

44 132 239 253 368 524 706 836 1041 1207 1338 1397

45 133 240 254 369 525 707 837 1042 1208 1339 1398

46 134 241 255 370 526 708 838 1043 1209 1340 1399

47 135 242 256 371 527 709 839 1044 1210 1341 1400

48 136 163 257 372 528 710 840 1045 1211 1342 1401

49 137 164 258 373 529 711 841 1046 1212 1343 1402

50 138 165 259 374 530 712 842 1047 1213 1344 1403

51 139 166 260 375 531 713 843 1048 1214 1345 1404

52 140 167 261 376 532 714 844 1049 1135 1346 1405

53 141 168 262 377 533 715 845 1050 1136 1347 1406

54 142 169 263 378 534 716 846 1051 1137 1348 1407

55 143 170 264 379 535 717 847 1052 1138 1349 1408

56 144 171 265 380 536 718 848 973 1139 1350 1409

57 145 172 266 381 537 719 849 974 1140 1351 1410

58 146 173 267 382 538 720 850 975 1141 1352 1411

59 147 174 268 383 539 721 851 976 1142 1353 1412

60 148 175 269 384 540 722 852 977 1143 1354 1413

61 149 176 270 385 541 723 853 978 1144 1355 1414

62 150 177 271 386 542 724 854 979 1145 1356 1415

63 151 178 272 387 543 725 855 980 1146 1357 1416

64 152 179 273 388 544 726 856 981 1147 1358 1417

65 153 180 274 389 545 727 857 982 1148 1359 1418

66 154 181 275 390 546 728 858 983 1149 1360 1419

67 155 182 276 391 547 649 859 984 1150 1361 1420

68 156 183 277 392 548 650 860 985 1151 1362 1421

69 157 184 278 393 549 651 861 986 1152 1363 1422

70 158 185 279 394 550 652 862 987 1153 1364 1423

71 159 186 280 395 551 653 863 988 1154 1365 1424

72 160 187 281 396 552 654 864 989 1155 1366 1425

73 161 188 282 397 553 655 865 990 1156 1367 1426

74 82 189 283 398 554 656 866 991 1157 1368 1427

75 83 190 284 399 555 657 867 992 1158 1369 1428

76 84 191 285 400 556 658 868 993 1159 1370 1429

77 85 192 286 401 557 659 869 994 1160 1371 1430

78 86 193 287 402 558 660 870 995 1161 1372 1431

79 87 194 288 403 559 661 871 996 1162 1373 1432

80 88 195 289 404 560 662 872 997 1163 1374 1433

1 89 196 290 325 561 663 873 998 1164 1375 1434

2 90 197 291 326 562 664 874 999 1165 1376 1435

3 91 198 292 327 563 665 875 1000 1166 1297 1436

4 92 199 293 328 564 666 876 1001 1167 1298 1437

5 93 200 294 329 565 667 877 1002 1168 1299 1438

6 94 201 295 330 566 668 878 1003 1169 1300 1439

7 95 202 296 331 487 669 879 1004 1170 1301 1440

8 96 203 297 332 488 670 880 1005 1171 1302 1441

9 97 204 298 333 489 671 881 1006 1172 1303 1442

10 98 205 299 334 490 672 882 1007 1173 1304 1443

11 99 206 300 335 491 673 883 1008 1174 1305 1444

12 100 207 301 336 492 674 884 1009 1175 1306 1445

13 101 208 302 337 493 675 885 1010 1176 1307 1446

15 148 220 319 424 616 785 943 1063 1252 1418 1527

67 113 241 281 448 574 807 914 1090 1234 1453 1459

55 129 200 286 423 601 757 961 1084 1230 1388 1528

68 105 208 244 437 578 745 929 1123 1254 1405 1473

39 117 191 267 418 583 758 932 1131 1228 1384 1538

14 83 238 266 416 621 809 952 1114 1220 1434 1510

28 111 206 260 449 644 784 939 1081 1282 1383 1461

9 81 209 299 483 643 798 965 1129 1247 1399 1483

7 152 229 307 421 637 779 915 1132 1227 1455 1467

40 125 216 290 414 596 777 903 1072 1259 1387 1533

74 160 242 257 456 604 730 916 1059 1269 1433 1530

12 149 192 305 463 587 764 967 1085 1275 1381 1491

5 94 180 308 465 634 782 942 1115 1286 1377 1535

47 159 193 248 443 602 775 956 1103 1243 1422 1525

54 131 164 315 406 605 737 937 1116 1283 1395 1520

56 82 219 261 482 625 744 935 1087 1287 1430 1511

34 104 233 291 426 612 746 968 1062 1222 1419 1480

77 88 214 279 480 638 804 922 1076 1239 1444 1501

73 154 212 292 472 588 767 940 1057 1218 1429 1537

17 151 165 263 454 576 763 933 1055 1268 1401 1476

71 112 199 318 419 589 787 895 1088 1217 1432 1485

63 156 217 252 479 640 761 902 1122 1233 1454 1512

45 146 210 313 431 615 753 904 1060 1289 1438 1489

10 141 172 311 441 629 735 962 1133 1221 1424 1494

70 132 179 264 447 610 780 925 1095 1267 1421 1532

22 101 181 298 458 608 760 921 1102 1295 1382 1475

32 122 239 316 415 641 792 945 1104 1215 1426 1474

38 158 202 309 455 595 802 919 1082 1256 1416 1468

49 97 198 271 459 613 808 911 1125 1229 1411 1507

6 106 222 278 474 606 739 893 1121 1264 1402 1515

46 133 196 280 411 568 776 938 1065 1253 1451 1498

50 110 188 258 470 575 736 918 1119 1278 1392 1465

65 115 170 301 440 577 740 950 1111 1263 1428 1513

2 153 215 297 469 635 755 960 1093 1235 1447 1516

61 96 195 321 485 598 772 966 1058 1266 1456 1536

31 95 227 295 461 594 751 897 1118 1288 1403 1523

60 89 237 287 473 618 801 934 1070 1272 1380 1469

76 128 163 269 439 592 750 894 1080 1258 1385 1499

52 136 174 314 467 584 766 898 1086 1255 1423 1487

64 119 211 294 405 646 742 913 1097 1216 1446 1500

30 86 171 246 428 611 754 930 1054 1260 1445 1471

58 134 175 256 481 591 800 909 1094 1250 1439 1526

0 137 190 262 436 623 748 959 1098 1245 1398 1460

19 157 207 273 425 633 729 908 1113 1236 1406 1521

72 144 186 310 450 639 789 924 1130 1285 1389 1519

27 90 236 270 435 570 762 900 1109 1226 1436 1472

16 120 185 274 407 607 797 912 1079 1262 1404 1506

41 108 201 289 438 647 786 958 1108 1281 1407 1524

26 121 177 306 460 571 731 906 1124 1290 1427 1517

78 92 189 285 444 586 796 891 1100 1237 1414 1479

29 147 235 255 430 603 768 947 1112 1294 1440 1486

51 161 183 284 427 582 799 920 1078 1219 1390 1488

35 142 162 300 468 632 741 955 1106 1257 1378 1466

21 140 224 276 432 581 805 944 1053 1280 1391 1509

18 93 197 288 422 597 791 969 1067 1279 1442 1505

59 127 232 254 417 573 788 954 1120 1273 1417 1529

23 145 221 282 408 585 749 926 1075 1232 1431 1503

13 138 166 243 457 631 793 957 1064 1240 1412 1495

8 100 231 323 478 579 783 899 1089 1223 1410 1477

79 107 203 296 434 567 778 963 1074 1270 1443 1522

48 109 234 251 453 620 769 949 1126 1238 1397 1502

69 87 176 320 462 593 738 946 1077 1241 1415 1534

25 130 240 265 409 628 759 907 1099 1261 1408 1464

44 126 226 250 466 617 795 951 1083 1248 1450 1470

53 150 223 302 471 642 734 941 1069 1274 1457 1481

80 124 184 253 429 627 743 936 1066 1224 1379 1518

57 116 228 275 452 599 770 927 1107 1292 1437 1478

62 98 218 259 451 630 747 896 1071 1225 1400 1482

20 143 213 245 445 572 752 917 1061 1276 1396 1497

43 123 204 322 484 636 790 953 1056 1251 1420 1514

42 155 173 283 412 622 733 892 1127 1265 1394 1493

36 85 194 247 475 619 732 901 1096 1246 1386 1463

75 91 230 317 442 580 806 928 1117 1244 1448 1492

3 102 169 312 410 624 765 905 1073 1277 1413 1508

66 139 178 303 413 614 773 910 1092 1231 1393 1484

33 99 205 272 433 609 756 948 1101 1249 1425 1496

1 103 182 293 420 600 803 971 1128 1242 1435 1462

4 118 187 249 446 569 771 970 1105 1284 1441 1490

24 135 225 268 476 590 774 964 1110 1291 1452 1458

11 114 168 277 464 626 794 923 1068 1293 1409 1531

37 84 167 304 477 645 781 931 1091 1271 1449 1504

16 149 221 320 395 502 707 838 977 1204 1528 1592

38 85 168 305 402 553 663 852 1022 1168 1505 1600

68 114 242 282 332 520 727 874 1038 1212 1460 1590

56 130 201 287 338 488 713 858 1014 1202 1529 1585

69 106 209 245 349 491 710 844 1026 1197 1474 1576

40 118 192 268 386 511 671 841 992 1188 1459 1545

15 84 239 267 346 498 715 882 1020 1157 1511 1566

29 112 207 261 350 524 705 846 972 1178 1462 1602

10 81 210 300 365 554 700 836 981 1214 1484 1541

8 153 230 308 382 542 691 831 1034 1153 1468 1550

41 126 217 291 361 555 660 822 989 1162 1534 1577

75 161 163 258 331 526 681 871 978 1189 1531 1554

13 150 193 306 360 501 717 812 1003 1166 1492 1559

6 95 181 309 376 515 656 848 988 1171 1536 1597

48 160 194 249 352 496 665 867 1040 1209 1526 1540

55 132 165 316 364 494 692 876 991 1152 1521 1619

57 83 220 262 330 527 669 823 1013 1151 1512 1613

35 105 234 292 358 561 674 880 997 1145 1481 1572

78 89 215 280 324 499 712 885 983 1184 1502 1580

74 155 213 293 399 492 655 843 980 1192 1538 1563

18 152 166 264 372 534 654 866 1021 1175 1477 1610

72 113 200 319 327 541 728 865 985 1142 1486 1578

64 157 218 253 396 543 687 859 975 1190 1513 1581

46 147 211 314 341 521 695 818 1050 1193 1490 1601

11 142 173 312 326 564 678 826 1041 1213 1495 1588

71 133 180 265 378 560 725 889 1010 1200 1533 1614

23 102 182 299 329 504 693 856 1031 1146 1476 1564

33 123 240 317 351 558 696 824 987 1176 1475 1552

39 159 203 310 335 550 716 827 1006 1164 1469 1565

50 98 199 272 401 532 703 847 1015 1177 1508 1616

7 107 223 279 398 497 649 834 1042 1148 1516 1591

47 134 197 281 359 557 679 860 1019 1203 1499 1605

51 111 189 259 403 509 667 890 1024 1137 1466 1586

66 116 171 302 393 519 680 878 982 1198 1514 1584

3 154 216 298 388 525 651 811 1005 1196 1517 1617

62 97 196 322 379 536 706 862 1004 1149 1537 1571

32 96 228 296 348 493 720 837 998 1183 1524 1589

61 90 238 288 369 533 701 851 1037 1201 1470 1582

77 129 164 270 325 537 699 832 1045 1194 1500 1544

53 137 175 315 344 552 652 830 1028 1156 1488 1551

65 120 212 295 353 489 686 863 995 1163 1501 1553

31 87 172 247 380 548 704 817 1043 1165 1472 1611

59 135 176 257 357 518 697 835 1046 1143 1527 1574

0 138 191 263 362 547 659 828 986 1186 1461 1570

20 158 208 274 400 563 666 870 973 1182 1522 1594

73 145 187 311 343 539 668 877 999 1206 1520 1568

28 91 237 271 342 551 726 879 1029 1180 1473 1560

17 121 186 275 336 517 689 857 1017 1172 1507 1542

42 109 202 290 375 545 685 820 1030 1154 1525 1587

27 122 178 307 383 486 709 816 1001 1199 1518 1567

79 93 190 286 366 506 683 840 976 1179 1480 1599

30 148 236 256 333 559 675 814 990 1211 1487 1609

52 82 184 285 381 514 657 886 1051 1141 1489 1615

36 143 162 301 384 503 702 868 1049 1147 1467 1546

22 141 225 277 404 528 682 833 1002 1158 1510 1583

19 94 198 289 391 513 714 813 1036 1195 1506 1543

60 128 233 255 337 565 724 845 974 1155 1530 1547

24 146 222 283 367 516 650 855 1047 1159 1504 1562

14 139 167 243 355 538 661 861 1009 1174 1496 1579

9 101 232 244 368 522 698 872 1016 1191 1478 1558

80 108 204 297 339 508 658 829 1018 1170 1523 1608

49 110 235 252 394 505 662 869 996 1140 1503 1557

70 88 177 321 328 546 677 873 1039 1169 1535 1573

26 131 241 266 389 510 694 888 1035 1185 1465 1549

45 127 227 251 387 500 673 825 979 1161 1471 1561

54 151 224 303 340 495 723 884 1033 1173 1482 1607

1 125 185 254 374 566 672 854 1025 1139 1519 1555

58 117 229 276 392 535 688 883 1007 1167 1479 1539

63 99 219 260 385 556 664 819 1052 1134 1483 1596

21 144 214 246 347 512 676 875 1032 1208 1498 1569

44 124 205 323 354 531 722 887 984 1181 1515 1604

43 156 174 284 356 540 670 853 994 1136 1494 1593

37 86 195 248 334 487 648 881 1000 1205 1464 1618

76 92 231 318 377 544 711 810 1011 1150 1493 1603

4 103 170 313 373 549 684 842 1048 1135 1509 1575

67 140 179 304 397 507 719 815 1008 1187 1485 1606

34 100 206 273 371 530 708 850 1012 1138 1497 1548

2 104 183 294 363 529 653 839 1027 1160 1463 1612

5 119 188 250 345 523 718 864 1044 1144 1491 1598

25 136 226 269 390 562 690 849 1023 1210 1458 1595

12 115 169 278 370 490 721 821 993 1207 1532 1556

17 150 222 321 426 618 787 945 1065 1254 1593 1633

13 116 170 279 466 628 796 925 1070 1295 1557 1641

39 86 169 306 479 647 783 933 1093 1273 1601 1631

69 115 163 283 450 576 809 916 1092 1236 1591 1626

57 131 202 288 425 603 759 963 1086 1232 1586 1697

70 107 210 246 439 580 747 931 1125 1256 1577 1666

41 119 193 269 420 585 760 934 1133 1230 1546 1687

16 85 240 268 418 623 731 954 1116 1222 1567 1643

30 113 208 262 451 646 786 941 1083 1284 1603 1662

11 81 211 301 485 645 800 967 1131 1249 1542 1671

9 154 231 309 423 639 781 917 1054 1229 1551 1698

42 127 218 292 416 598 779 905 1074 1261 1578 1675

76 82 164 259 458 606 732 918 1061 1271 1555 1680

14 151 194 307 465 589 766 969 1087 1277 1560 1638

7 96 182 310 467 636 784 944 1117 1288 1598 1661

49 161 195 250 445 604 777 958 1105 1245 1541 1660

56 133 166 317 408 607 739 939 1118 1285 1540 1654

58 84 221 263 484 627 746 937 1089 1289 1614 1693

36 106 235 293 428 614 748 970 1064 1224 1573 1621

79 90 216 281 482 640 806 924 1078 1241 1581 1684

75 156 214 294 474 590 769 942 1059 1220 1564 1651

19 153 167 265 456 578 765 935 1057 1270 1611 1699

73 114 201 320 421 591 789 897 1090 1219 1579 1622

65 158 219 254 481 642 763 904 1124 1235 1582 1642

47 148 212 315 433 617 755 906 1062 1291 1602 1629

12 143 174 313 443 631 737 964 1055 1223 1589 1655

72 134 181 266 449 612 782 927 1097 1269 1615 1685

24 103 183 300 460 610 762 923 1104 1217 1565 1673

34 124 241 318 417 643 794 947 1106 1215 1553 1686

40 160 204 311 457 597 804 921 1084 1258 1566 1657

51 99 200 273 461 615 730 913 1127 1231 1617 1632

8 108 224 280 476 608 741 895 1123 1266 1592 1646

48 135 198 282 413 570 778 940 1067 1255 1606 1627

52 112 190 260 472 577 738 920 1121 1280 1587 1625

67 117 172 303 442 579 742 952 1113 1265 1585 1658

4 155 217 299 471 637 757 962 1095 1237 1618 1692

63 98 197 323 407 600 774 968 1060 1268 1572 1630

33 97 229 297 463 596 753 899 1120 1290 1590 1623

62 91 239 289 475 620 803 936 1072 1274 1583 1665

78 130 165 271 441 594 752 896 1082 1260 1545 1672

54 138 176 316 469 586 768 900 1088 1257 1552 1674

66 121 213 296 405 568 744 915 1099 1218 1554 1652

32 88 173 248 430 613 756 932 1056 1262 1612 1695

60 136 177 258 483 593 802 911 1096 1252 1575 1691

0 139 192 264 438 625 750 961 1100 1247 1571 1635

21 159 209 275 427 635 729 910 1115 1238 1595 1689

74 146 188 312 452 641 791 926 1132 1287 1569 1681

29 92 238 272 437 572 764 902 1111 1228 1561 1663

18 122 187 276 409 609 799 914 1081 1264 1543 1628

43 110 203 291 440 569 788 960 1110 1283 1588 1688

28 123 179 308 462 573 733 908 1126 1292 1568 1640

80 94 191 287 446 588 798 891 1102 1239 1600 1650

31 149 237 257 432 605 770 949 1114 1216 1610 1656

53 83 185 286 429 584 801 922 1080 1221 1616 1667

37 144 162 302 470 634 743 957 1108 1259 1547 1624

23 142 226 278 434 583 807 946 1053 1282 1584 1664

20 95 199 290 424 599 793 971 1069 1281 1544 1668

61 129 234 256 419 575 790 956 1122 1275 1548 1683

25 147 223 284 410 587 751 928 1077 1234 1563 1700

15 140 168 243 459 633 795 959 1066 1242 1580 1679

10 102 233 245 480 581 785 901 1091 1225 1559 1649

1 109 205 298 436 567 780 965 1076 1272 1609 1678

50 111 236 253 455 622 771 951 1128 1240 1558 1694

71 89 178 322 464 595 740 948 1079 1243 1574 1670

27 132 242 267 411 630 761 909 1101 1263 1550 1682

46 128 228 252 468 619 797 953 1085 1250 1562 1648

55 152 225 304 473 644 736 943 1071 1276 1608 1676

2 126 186 255 431 629 745 938 1068 1226 1556 1620

59 118 230 277 454 601 772 929 1109 1294 1539 1637

64 100 220 261 453 632 749 898 1073 1227 1597 1690

22 145 215 247 447 574 754 919 1063 1278 1570 1645

45 125 206 244 406 638 792 955 1058 1253 1605 1634

44 157 175 285 414 624 735 894 1129 1267 1594 1659

38 87 196 249 477 621 734 903 1098 1248 1619 1644

77 93 232 319 444 582 808 930 1119 1246 1604 1696

5 104 171 314 412 626 767 907 1075 1279 1576 1647

68 141 180 305 415 616 775 912 1094 1233 1607 1669

35 101 207 274 435 611 758 950 1103 1251 1549 1653

3 105 184 295 422 602 805 893 1130 1244 1613 1639

6 120 189 251 448 571 773 892 1107 1286 1599 1636

26 137 227 270 478 592 776 966 1112 1293 1596 1677

18 151 223 322 397 504 709 840 979 1206 1634 1781

27 138 228 271 392 564 692 851 1025 1212 1678 1734

14 117 171 280 372 492 723 823 995 1209 1642 1755

40 87 170 307 404 555 665 854 1024 1170 1632 1711

70 116 164 284 334 522 649 876 1040 1214 1627 1730

58 132 203 289 340 490 715 860 1016 1204 1698 1739

71 108 211 247 351 493 712 846 1028 1199 1667 1766

42 120 194 270 388 513 673 843 994 1190 1688 1743

17 86 241 269 348 500 717 884 1022 1159 1644 1748

31 114 209 263 352 526 707 848 972 1180 1663 1706

12 81 212 302 367 556 702 838 983 1136 1672 1729

10 155 232 310 384 544 693 833 1036 1155 1699 1728

43 128 219 293 363 557 662 824 991 1164 1676 1722

77 83 165 260 333 528 683 873 980 1191 1681 1761

15 152 195 308 362 503 719 814 1005 1168 1639 1769

8 97 183 311 378 517 658 850 990 1173 1662 1752

50 82 196 251 354 498 667 869 1042 1211 1661 1719

57 134 167 318 366 496 694 878 993 1154 1655 1767

59 85 222 264 332 529 671 825 1015 1153 1694 1770

37 107 236 294 360 563 676 882 999 1147 1622 1710

80 91 217 282 324 501 714 887 985 1186 1685 1777

76 157 215 295 401 494 657 845 982 1194 1652 1723

20 154 168 266 374 536 656 868 1023 1177 1700 1753

74 115 202 321 329 543 650 867 987 1144 1623 1741

66 159 220 255 398 545 689 861 977 1192 1643 1754

48 149 213 316 343 523 697 820 1052 1195 1630 1725

13 144 175 314 328 566 680 828 1043 1135 1656 1780

73 135 182 267 380 562 727 811 1012 1202 1686 1714

25 104 184 301 331 506 695 858 1033 1148 1674 1775

35 125 242 319 353 560 698 826 989 1178 1687 1773

41 161 205 312 337 552 718 829 1008 1166 1658 1726

52 100 201 274 403 534 705 849 1017 1179 1633 1760

9 109 225 281 400 499 651 836 1044 1150 1647 1778

49 136 199 283 361 559 681 862 1021 1205 1628 1771

53 113 191 261 325 511 669 812 1026 1139 1626 1733

68 118 173 304 395 521 682 880 984 1200 1659 1740

5 156 218 300 390 527 653 813 1007 1198 1693 1742

64 99 198 244 381 538 708 864 1006 1151 1631 1720

34 98 230 298 350 495 722 839 1000 1185 1624 1763

63 92 240 290 371 535 703 853 1039 1203 1666 1759

79 131 166 272 327 539 701 834 1047 1196 1673 1703

55 139 177 317 346 554 654 832 1030 1158 1675 1757

67 122 214 297 355 491 688 865 997 1165 1653 1749

33 89 174 249 382 550 706 819 1045 1167 1696 1731

61 137 178 259 359 520 699 837 1048 1145 1692 1776

0 140 193 265 364 549 661 830 988 1188 1636 1756

22 160 210 276 402 565 668 872 975 1184 1690 1708

75 147 189 313 345 541 670 879 1001 1208 1682 1718

30 93 239 273 344 553 728 881 1031 1182 1664 1724

19 123 188 277 338 519 691 859 1019 1174 1629 1735

44 111 204 292 377 547 687 822 1032 1156 1689 1772

29 124 180 309 385 486 711 818 1003 1201 1641 1732

1 95 192 288 368 508 685 842 978 1181 1651 1736

32 150 238 258 335 561 677 816 992 1213 1657 1751

54 84 186 287 383 516 659 888 973 1143 1668 1768

38 145 162 303 386 505 704 870 1051 1149 1625 1747

24 143 227 279 326 530 684 835 1004 1160 1665 1717

21 96 200 291 393 515 716 815 1038 1197 1669 1746

62 130 235 257 339 487 726 847 976 1157 1684 1762

26 148 224 285 369 518 652 857 1049 1161 1621 1738

16 141 169 243 357 540 663 863 1011 1176 1680 1750

11 103 234 246 370 524 700 874 1018 1193 1650 1716

2 110 206 299 341 510 660 831 1020 1172 1679 1744

51 112 237 254 396 507 664 871 998 1142 1695 1701

72 90 179 323 330 548 679 875 1041 1171 1671 1705

28 133 163 268 391 512 696 890 1037 1187 1683 1758

47 129 229 253 389 502 675 827 981 1163 1649 1713

56 153 226 305 342 497 725 886 1035 1175 1677 1702

3 127 187 256 376 488 674 856 1027 1141 1620 1727

60 119 231 278 394 537 690 885 1009 1169 1638 1712

65 101 221 262 387 558 666 821 974 1134 1691 1764

23 146 216 248 349 514 678 877 1034 1210 1646 1715

46 126 207 245 356 533 724 889 986 1183 1635 1737

45 158 176 286 358 542 672 855 996 1138 1660 1721

39 88 197 250 336 489 648 883 1002 1207 1645 1707

78 94 233 320 379 546 713 810 1013 1152 1697 1704

6 105 172 315 375 551 686 844 1050 1137 1648 1745

69 142 181 306 399 509 721 817 1010 1189 1670 1709

36 102 208 275 373 532 710 852 1014 1140 1654 1779

4 106 185 296 365 531 655 841 1029 1162 1640 1774

7 121 190 252 347 525 720 866 1046 1146 1637 1765

19 152 224 323 428 620 789 947 1067 1256 1702 1788

8 122 191 253 450 573 775 894 1109 1288 1766 1801

28 139 229 272 480 594 778 968 1114 1295 1735 1862

15 118 172 281 468 630 798 927 1072 1217 1756 1860

41 88 171 308 481 569 785 935 1095 1275 1712 1813

71 117 165 285 452 578 731 918 1094 1238 1731 1847

59 133 204 290 427 605 761 965 1088 1234 1740 1785

72 109 212 248 441 582 749 933 1127 1258 1767 1858

43 121 195 271 422 587 762 936 1055 1232 1744 1820

18 87 242 270 420 625 733 956 1118 1224 1749 1827

32 115 210 264 453 568 788 943 1085 1286 1707 1829

13 81 213 303 407 647 802 969 1133 1251 1730 1807

11 156 233 311 425 641 783 919 1056 1231 1729 1850

44 129 220 294 418 600 781 907 1076 1263 1723 1846

78 84 166 261 460 608 734 920 1063 1273 1762 1790

16 153 196 309 467 591 768 971 1089 1279 1770 1844

9 98 184 312 469 638 786 946 1119 1290 1753 1836

51 83 197 252 447 606 779 960 1107 1247 1720 1818

58 135 168 319 410 609 741 941 1120 1287 1768 1783

60 86 223 265 406 629 748 939 1091 1291 1771 1843

38 108 237 295 430 616 750 892 1066 1226 1711 1795

1 92 218 283 484 642 808 926 1080 1243 1778 1805

77 158 216 296 476 592 771 944 1061 1222 1724 1811

21 155 169 267 458 580 767 937 1059 1272 1754 1822

75 116 203 322 423 593 791 899 1092 1221 1742 1859

67 160 221 256 483 644 765 906 1126 1237 1755 1819

49 150 214 317 435 619 757 908 1064 1293 1726 1823

14 145 176 315 445 633 739 966 1057 1225 1781 1838

74 136 183 268 451 614 784 929 1099 1271 1715 1855

26 105 185 302 462 612 764 925 1106 1219 1776 1834

36 126 163 320 419 645 796 949 1108 1215 1774 1804

42 82 206 313 459 599 806 923 1086 1260 1727 1833

53 101 202 275 463 617 732 915 1129 1233 1761 1849

10 110 226 282 478 610 743 897 1125 1268 1779 1825

50 137 200 284 415 572 780 942 1069 1257 1772 1837

54 114 192 262 474 579 740 922 1123 1282 1734 1803

69 119 174 305 444 581 744 954 1115 1267 1741 1831

6 157 219 301 473 639 759 964 1097 1239 1743 1782

65 100 199 245 409 602 776 970 1062 1270 1721 1792

35 99 231 299 465 598 755 901 1122 1292 1764 1845

64 93 241 291 477 622 805 938 1074 1276 1760 1800

80 132 167 273 443 596 754 898 1084 1262 1704 1789

56 140 178 318 471 588 770 902 1090 1259 1758 1814

68 123 215 298 405 570 746 917 1101 1220 1750 1799

34 90 175 250 432 615 758 934 1058 1264 1732 1851

62 138 179 260 485 595 804 913 1098 1254 1777 1802

0 141 194 266 440 627 752 963 1102 1249 1757 1824

23 161 211 277 429 637 729 912 1117 1240 1709 1808

76 148 190 314 454 643 793 928 1054 1289 1719 1794

31 94 240 274 439 574 766 904 1113 1230 1725 1791

20 124 189 278 411 611 801 916 1083 1266 1736 1832

45 112 205 293 442 571 790 962 1112 1285 1773 1796

30 125 181 310 464 575 735 910 1128 1294 1733 1786

2 96 193 289 448 590 800 891 1104 1241 1737 1861

33 151 239 259 434 607 772 951 1116 1218 1752 1852

55 85 187 288 431 586 803 924 1082 1223 1769 1821

39 146 162 304 472 636 745 959 1110 1261 1748 1842

25 144 228 280 436 585 809 948 1053 1284 1718 1798

22 97 201 292 426 601 795 893 1071 1283 1747 1817

63 131 236 258 421 577 792 958 1124 1277 1763 1826

27 149 225 286 412 589 753 930 1079 1236 1739 1853

17 142 170 243 461 635 797 961 1068 1244 1751 1830

12 104 235 247 482 583 787 903 1093 1227 1717 1835

3 111 207 300 438 567 782 967 1078 1274 1745 1793

52 113 238 255 457 624 773 953 1130 1242 1701 1816

73 91 180 244 466 597 742 950 1081 1245 1706 1815

29 134 164 269 413 632 763 911 1103 1265 1759 1809

48 130 230 254 470 621 799 955 1087 1252 1714 1848

57 154 227 306 475 646 738 945 1073 1278 1703 1856

4 128 188 257 433 631 747 940 1070 1228 1728 1839

61 120 232 279 456 603 774 931 1111 1216 1713 1806

66 102 222 263 455 634 751 900 1075 1229 1765 1854

24 147 217 249 449 576 756 921 1065 1280 1716 1857

47 127 208 246 408 640 794 957 1060 1255 1738 1797

46 159 177 287 416 626 737 896 1131 1269 1722 1784

40 89 198 251 479 623 736 905 1100 1250 1708 1810

79 95 234 321 446 584 730 932 1121 1248 1705 1840

7 106 173 316 414 628 769 909 1077 1281 1746 1828

70 143 182 307 417 618 777 914 1096 1235 1710 1841

37 103 209 276 437 613 760 952 1105 1253 1780 1812

5 107 186 297 424 604 807 895 1132 1246 1775 1787

20 153 225 244 399 506 711 842 981 1208 1789 1863

6 108 187 298 367 533 657 843 1031 1164 1788 1865

9 123 192 254 349 527 722 868 1048 1148 1802 1866

29 140 230 273 394 566 694 853 1027 1214 1783 1867

16 119 173 282 374 494 725 825 997 1211 1861 1868

42 89 172 309 326 557 667 856 1026 1172 1814 1869

72 118 166 286 336 524 651 878 1042 1136 1848 1870

60 134 205 291 342 492 717 862 1018 1206 1786 1871

73 110 213 249 353 495 714 848 1030 1201 1859 1872

44 122 196 272 390 515 675 845 996 1192 1821 1873

19 88 163 271 350 502 719 886 1024 1161 1828 1874

33 116 211 265 354 528 709 850 972 1182 1830 1875

14 81 214 304 369 558 704 840 985 1138 1808 1876

12 157 234 312 386 546 695 835 1038 1157 1851 1877

45 130 221 295 365 559 664 826 993 1166 1847 1878

79 85 167 262 335 530 685 875 982 1193 1791 1879

17 154 197 310 364 505 721 816 1007 1170 1845 1880

10 99 185 313 380 519 660 852 992 1175 1837 1881

52 84 198 253 356 500 669 871 1044 1213 1819 1882

59 136 169 320 368 498 696 880 995 1156 1784 1883

61 87 224 266 334 531 673 827 1017 1155 1844 1884

39 109 238 296 362 565 678 884 1001 1149 1796 1885

2 93 219 284 324 503 716 889 987 1188 1806 1886

78 159 217 297 403 496 659 847 984 1196 1812 1887

22 156 170 268 376 538 658 870 1025 1179 1823 1888

76 117 204 323 331 545 652 869 989 1146 1860 1889

68 161 222 257 400 547 691 863 979 1194 1820 1890

50 151 215 318 345 525 699 822 974 1197 1824 1891

15 146 177 316 330 488 682 830 1045 1137 1839 1892

75 137 184 269 382 564 649 813 1014 1204 1856 1893

27 106 186 303 333 508 697 860 1035 1150 1835 1894

37 127 164 321 355 562 700 828 991 1180 1805 1895

43 83 207 314 339 554 720 831 1010 1168 1834 1896

54 102 203 276 325 536 707 851 1019 1181 1850 1897

11 111 227 283 402 501 653 838 1046 1152 1826 1898

51 138 201 285 363 561 683 864 1023 1207 1838 1899

55 115 193 263 327 513 671 814 1028 1141 1804 1900

70 120 175 306 397 523 684 882 986 1202 1832 1901

7 158 220 302 392 529 655 815 1009 1200 1782 1902

66 101 200 246 383 540 710 866 1008 1153 1793 1903

36 100 232 300 352 497 724 841 1002 1187 1846 1904

65 94 242 292 373 537 705 855 1041 1205 1801 1905

1 133 168 274 329 541 703 836 1049 1198 1790 1906

57 141 179 319 348 556 656 834 1032 1160 1815 1907

69 124 216 299 357 493 690 867 999 1167 1800 1908

35 91 176 251 384 552 708 821 1047 1169 1852 1909

63 139 180 261 361 522 701 839 1050 1147 1803 1910

0 142 195 267 366 551 663 832 990 1190 1825 1911

24 82 212 278 404 487 670 874 977 1186 1809 1912

77 149 191 315 347 543 672 881 1003 1210 1795 1913

32 95 241 275 346 555 650 883 1033 1184 1792 1914

21 125 190 279 340 521 693 861 1021 1176 1833 1915

46 113 206 294 379 549 689 824 1034 1158 1797 1916

31 126 182 311 387 486 713 820 1005 1203 1787 1917

3 97 194 290 370 510 687 844 980 1183 1862 1918

34 152 240 260 337 563 679 818 994 1135 1853 1919

56 86 188 289 385 518 661 890 975 1145 1822 1920

40 147 162 305 388 507 706 872 973 1151 1843 1921

26 145 229 281 328 532 686 837 1006 1162 1799 1922

23 98 202 293 395 517 718 817 1040 1199 1818 1923

64 132 237 259 341 489 728 849 978 1159 1827 1924

28 150 226 287 371 520 654 859 1051 1163 1854 1925

18 143 171 243 359 542 665 865 1013 1178 1831 1926

13 105 236 248 372 526 702 876 1020 1195 1836 1927

4 112 208 301 343 512 662 833 1022 1174 1794 1928

53 114 239 256 398 509 666 873 1000 1144 1817 1929

74 92 181 245 332 550 681 877 1043 1173 1816 1930

30 135 165 270 393 514 698 812 1039 1189 1810 1931

49 131 231 255 391 504 677 829 983 1165 1849 1932

58 155 228 307 344 499 727 888 1037 1177 1857 1933

5 129 189 258 378 490 676 858 1029 1143 1840 1934

62 121 233 280 396 539 692 887 1011 1171 1807 1935

67 103 223 264 389 560 668 823 976 1134 1855 1936

25 148 218 250 351 516 680 879 1036 1212 1858 1937

48 128 209 247 358 535 726 811 988 1185 1798 1938

47 160 178 288 360 544 674 857 998 1140 1785 1939

41 90 199 252 338 491 648 885 1004 1209 1811 1940

80 96 235 322 381 548 715 810 1015 1154 1841 1941

8 107 174 317 377 553 688 846 1052 1139 1829 1942

71 144 183 308 401 511 723 819 1012 1191 1842 1943

38 104 210 277 375 534 712 854 1016 1142 1813 1864

21 154 226 245 430 622 791 949 1069 1258 1863

39 105 211 278 439 615 762 954 1107 1255 1865

7 109 188 299 426 606 809 897 1054 1248 1866

10 124 193 255 452 575 777 896 1111 1290 1867

30 141 231 274 482 596 780 970 1116 1217 1868

17 120 174 283 470 632 800 929 1074 1219 1869

43 90 173 310 483 571 787 937 1097 1277 1870

73 119 167 287 454 580 733 920 1096 1240 1871

61 135 206 292 429 607 763 967 1090 1236 1872

74 111 214 250 443 584 751 935 1129 1260 1873

45 123 197 273 424 589 764 938 1057 1234 1874

20 89 164 272 422 627 735 958 1120 1226 1875

34 117 212 266 455 570 790 945 1087 1288 1876

15 81 215 305 409 569 804 971 1055 1253 1877

13 158 235 313 427 643 785 921 1058 1233 1878

46 131 222 296 420 602 783 909 1078 1265 1879

80 86 168 263 462 610 736 922 1065 1275 1880

18 155 198 311 469 593 770 893 1091 1281 1881

11 100 186 314 471 640 788 948 1121 1292 1882

53 85 199 254 449 608 781 962 1109 1249 1883

60 137 170 321 412 611 743 943 1122 1289 1884

62 88 225 267 408 631 750 941 1093 1293 1885

40 110 239 297 432 618 752 894 1068 1228 1886

3 94 220 285 406 644 730 928 1082 1245 1887

79 160 218 298 478 594 773 946 1063 1224 1888

23 157 171 269 460 582 769 939 1061 1274 1889

77 118 205 244 425 595 793 901 1094 1223 1890

69 82 223 258 485 646 767 908 1128 1239 1891

51 152 216 319 437 621 759 910 1066 1295 1892

16 147 178 317 447 635 741 968 1059 1227 1893

76 138 185 270 453 616 786 931 1101 1273 1894

28 107 187 304 464 614 766 927 1108 1221 1895

38 128 165 322 421 647 798 951 1110 1215 1896

44 84 208 315 461 601 808 925 1088 1262 1897

55 103 204 277 465 619 734 917 1131 1235 1898

12 112 228 284 480 612 745 899 1127 1270 1899

52 139 202 286 417 574 782 944 1071 1259 1900

56 116 194 264 476 581 742 924 1125 1284 1901

71 121 176 307 446 583 746 956 1117 1269 1902

8 159 221 303 475 641 761 966 1099 1241 1903

67 102 201 247 411 604 778 892 1064 1272 1904

37 101 233 301 467 600 757 903 1124 1294 1905

66 95 163 293 479 624 807 940 1076 1278 1906

2 134 169 275 445 598 756 900 1086 1264 1907

58 142 180 320 473 590 772 904 1092 1261 1908

70 125 217 300 405 572 748 919 1103 1222 1909

36 92 177 252 434 617 760 936 1060 1266 1910

64 140 181 262 407 597 806 915 1100 1256 1911

0 143 196 268 442 629 754 965 1104 1251 1912

25 83 213 279 431 639 729 914 1119 1242 1913

78 150 192 316 456 645 795 930 1056 1291 1914

33 96 242 276 441 576 768 906 1115 1232 1915

22 126 191 280 413 613 803 918 1085 1268 1916

47 114 207 295 444 573 792 964 1114 1287 1917

32 127 183 312 466 577 737 912 1130 1216 1918

4 98 195 291 450 592 802 891 1106 1243 1919

35 153 241 261 436 609 774 953 1118 1220 1920

57 87 189 290 433 588 805 926 1084 1225 1921

41 148 162 306 474 638 747 961 1112 1263 1922

27 146 230 282 438 587 731 950 1053 1286 1923

24 99 203 294 428 603 797 895 1073 1285 1924

65 133 238 260 423 579 794 960 1126 1279 1925

29 151 227 288 414 591 755 932 1081 1238 1926

19 144 172 243 463 637 799 963 1070 1246 1927

14 106 237 249 484 585 789 905 1095 1229 1928

5 113 209 302 440 567 784 969 1080 1276 1929

54 115 240 257 459 626 775 955 1132 1244 1930

75 93 182 246 468 599 744 952 1083 1247 1931

31 136 166 271 415 634 765 913 1105 1267 1932

50 132 232 256 472 623 801 957 1089 1254 1933

59 156 229 308 477 568 740 947 1075 1280 1934

6 130 190 259 435 633 749 942 1072 1230 1935

63 122 234 281 458 605 776 933 1113 1218 1936

68 104 224 265 457 636 753 902 1077 1231 1937

26 149 219 251 451 578 758 923 1067 1282 1938

49 129 210 248 410 642 796 959 1062 1257 1939

48 161 179 289 418 628 739 898 1133 1271 1940

42 91 200 253 481 625 738 907 1102 1252 1941

1 97 236 323 448 586 732 934 1123 1250 1942

9 108 175 318 416 630 771 911 1079 1283 1943

72 145 184 309 419 620 779 916 1098 1237 1864

6. based on GRS abnormal LDPC (1944,487) sign indicating number (6), R=3/4

1944 486

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16 16 16 16 16 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 16

0 81 162 243 324 405 567 648 729 810 1053 1134 1215 1296 1458 1539

14 102 209 303 338 426 598 676 783 886 1131 1177 1267 1308 1490 1566

15 103 210 304 339 427 599 677 784 887 1132 1178 1268 1309 1491 1567

16 104 211 305 340 428 600 678 785 888 1133 1179 1269 1310 1492 1568

17 105 212 306 341 429 601 679 786 889 1054 1180 1270 1311 1493 1569

18 106 213 307 342 430 602 680 787 890 1055 1181 1271 1312 1494 1570

19 107 214 308 343 431 603 681 788 811 1056 1182 1272 1313 1495 1571

20 108 215 309 344 432 604 682 789 812 1057 1183 1273 1314 1496 1572

21 109 216 310 345 433 605 683 790 813 1058 1184 1274 1315 1497 1573

22 110 217 311 346 434 606 684 791 814 1059 1185 1275 1316 1498 1574

23 111 218 312 347 435 607 685 792 815 1060 1186 1276 1317 1499 1575

24 112 219 313 348 436 608 686 793 816 1061 1187 1277 1318 1500 1576

25 113 220 314 349 437 609 687 794 817 1062 1188 1278 1319 1501 1577

26 114 221 315 350 438 610 688 795 818 1063 1189 1279 1320 1502 1578

27 115 222 316 351 439 611 689 796 819 1064 1190 1280 1321 1503 1579

28 116 223 317 352 440 612 690 797 820 1065 1191 1281 1322 1504 1580

29 117 224 318 353 441 613 691 798 821 1066 1192 1282 1323 1505 1581

30 118 225 319 354 442 614 692 799 822 1067 1193 1283 1324 1506 1582

31 119 226 320 355 443 615 693 800 823 1068 1194 1284 1325 1507 1583

32 120 227 321 356 444 616 694 801 824 1069 1195 1285 1326 1508 1584

33 121 228 322 357 445 617 695 802 825 1070 1196 1286 1327 1509 1585

34 122 229 323 358 446 618 696 803 826 1071 1197 1287 1328 1510 1586

35 123 230 244 359 447 619 697 804 827 1072 1198 1288 1329 1511 1587

36 124 231 245 360 448 620 698 805 828 1073 1199 1289 1330 1512 1588

37 125 232 246 361 449 621 699 806 829 1074 1200 1290 1331 1513 1589

38 126 233 247 362 450 622 700 807 830 1075 1201 1291 1332 1514 1590

39 127 234 248 363 451 623 701 808 831 1076 1202 1292 1333 1515 1591

40 128 235 249 364 452 624 702 809 832 1077 1203 1293 1334 1516 1592

41 129 236 250 365 453 625 703 730 833 1078 1204 1294 1335 1517 1593

42 130 237 251 366 454 626 704 731 834 1079 1205 1295 1336 1518 1594

43 131 238 252 367 455 627 705 732 835 1080 1206 1216 1337 1519 1595

44 132 239 253 368 456 628 706 733 836 1081 1207 1217 1338 1520 1596

45 133 240 254 369 457 629 707 734 837 1082 1208 1218 1339 1521 1597

46 134 241 255 370 458 630 708 735 838 1083 1209 1219 1340 1522 1598

47 135 242 256 371 459 631 709 736 839 1084 1210 1220 1341 1523 1599

48 136 163 257 372 460 632 710 737 840 1085 1211 1221 1342 1524 1600

49 137 164 258 373 461 633 711 738 841 1086 1212 1222 1343 1525 1601

50 138 165 259 374 462 634 712 739 842 1087 1213 1223 1344 1526 1602

51 139 166 260 375 463 635 713 740 843 1088 1214 1224 1345 1527 1603

52 140 167 261 376 464 636 714 741 844 1089 1135 1225 1346 1528 1604

53 141 168 262 377 465 637 715 742 845 1090 1136 1226 1347 1529 1605

54 142 169 263 378 466 638 716 743 846 1091 1137 1227 1348 1530 1606

55 143 170 264 379 467 639 717 744 847 1092 1138 1228 1349 1531 1607

56 144 171 265 380 468 640 718 745 848 1093 1139 1229 1350 1532 1608

57 145 172 266 381 469 641 719 746 849 1094 1140 1230 1351 1533 1609

58 146 173 267 382 470 642 720 747 850 1095 1141 1231 1352 1534 1610

59 147 174 268 383 471 643 721 748 851 1096 1142 1232 1353 1535 1611

60 148 175 269 384 472 644 722 749 852 1097 1143 1233 1354 1536 1612

61 149 176 270 385 473 645 723 750 853 1098 1144 1234 1355 1537 1613

62 150 177 271 386 474 646 724 751 854 1099 1145 1235 1356 1538 1614

63 151 178 272 387 475 647 725 752 855 1100 1146 1236 1357 1459 1615

64 152 179 273 388 476 568 726 753 856 1101 1147 1237 1358 1460 1616

65 153 180 274 389 477 569 727 754 857 1102 1148 1238 1359 1461 1617

66 154 181 275 390 478 570 728 755 858 1103 1149 1239 1360 1462 1618

67 155 182 276 391 479 571 649 756 859 1104 1150 1240 1361 1463 1619

68 156 183 277 392 480 572 650 757 860 1105 1151 1241 1362 1464 1540

69 157 184 278 393 481 573 651 758 861 1106 1152 1242 1363 1465 1541

70 158 185 279 394 482 574 652 759 862 1107 1153 1243 1364 1466 1542

71 159 186 280 395 483 575 653 760 863 1108 1154 1244 1365 1467 1543

72 160 187 281 396 484 576 654 761 864 1109 1155 1245 1366 1468 1544

73 161 188 282 397 485 577 655 762 865 1110 1156 1246 1367 1469 1545

74 82 189 283 398 406 578 656 763 866 1111 1157 1247 1368 1470 1546

75 83 190 284 399 407 579 657 764 867 1112 1158 1248 1369 1471 1547

76 84 191 285 400 408 580 658 765 868 1113 1159 1249 1370 1472 1548

77 85 192 286 401 409 581 659 766 869 1114 1160 1250 1371 1473 1549

78 86 193 287 402 410 582 660 767 870 1115 1161 1251 1372 1474 1550

79 87 194 288 403 411 583 661 768 871 1116 1162 1252 1373 1475 1551

80 88 195 289 404 412 584 662 769 872 1117 1163 1253 1374 1476 1552

1 89 196 290 325 413 585 663 770 873 1118 1164 1254 1375 1477 1553

2 90 197 291 326 414 586 664 771 874 1119 1165 1255 1376 1478 1554

3 91 198 292 327 415 587 665 772 875 1120 1166 1256 1297 1479 1555

4 92 199 293 328 416 588 666 773 876 1121 1167 1257 1298 1480 1556

5 93 200 294 329 417 589 667 774 877 1122 1168 1258 1299 1481 1557

6 94 201 295 330 418 590 668 775 878 1123 1169 1259 1300 1482 1558

7 95 202 296 331 419 591 669 776 879 1124 1170 1260 1301 1483 1559

8 96 203 297 332 420 592 670 777 880 1125 1171 1261 1302 1484 1560

9 97 204 298 333 421 593 671 778 881 1126 1172 1262 1303 1485 1561

10 98 205 299 334 422 594 672 779 882 1127 1173 1263 1304 1486 1562

11 99 206 300 335 423 595 673 780 883 1128 1174 1264 1305 1487 1563

12 100 207 301 336 424 596 674 781 884 1129 1175 1265 1306 1488 1564

13 101 208 302 337 425 597 675 782 885 1130 1176 1266 1307 1489 1565

15 148 220 319 394 424 706 785 837 943 1203 1252 1356 1418 1591 1631

67 113 241 281 331 448 726 807 873 914 1211 1234 1322 1453 1589 1624

55 129 200 286 337 423 712 757 857 961 1201 1230 1372 1388 1584 1695

68 105 208 244 348 437 709 745 843 929 1196 1254 1321 1405 1575 1664

39 117 191 267 385 418 670 758 840 932 1187 1228 1337 1384 1544 1685

14 83 238 266 345 416 714 809 881 952 1156 1220 1313 1434 1565 1641

28 111 206 260 349 449 704 784 845 939 1177 1282 1325 1383 1601 1660

9 81 209 299 364 483 699 798 835 965 1213 1247 1371 1399 1540 1669

7 152 229 307 381 421 690 779 830 915 1152 1227 1319 1455 1549 1696

40 125 216 290 360 414 659 777 821 903 1161 1259 1296 1387 1576 1673

74 160 242 257 330 456 680 730 870 916 1188 1269 1360 1433 1553 1678

12 149 192 305 359 463 716 764 811 967 1165 1275 1333 1381 1558 1636

5 94 180 308 375 465 655 782 847 942 1170 1286 1368 1377 1596 1659

47 159 193 248 351 443 664 775 866 956 1208 1243 1357 1422 1619 1658

54 131 164 315 363 406 691 737 875 937 1151 1283 1302 1395 1618 1652

56 82 219 261 329 482 668 744 822 935 1150 1287 1367 1430 1612 1691

34 104 233 291 357 426 673 746 879 968 1144 1222 1339 1419 1571 1699

77 88 214 279 324 480 711 804 884 922 1183 1239 1370 1444 1579 1682

73 154 212 292 398 472 654 767 842 940 1191 1218 1312 1429 1562 1649

17 151 165 263 371 454 653 763 865 933 1174 1268 1376 1401 1609 1697

71 112 199 318 326 419 727 787 864 895 1141 1217 1362 1432 1577 1700

63 156 217 252 395 479 686 761 858 902 1189 1233 1359 1454 1580 1640

45 146 210 313 340 431 694 753 817 904 1192 1289 1320 1438 1600 1627

10 141 172 311 325 441 677 735 825 962 1212 1221 1364 1424 1587 1653

70 132 179 264 377 447 724 780 888 925 1199 1267 1354 1421 1613 1683

22 101 181 298 328 458 692 760 855 921 1145 1295 1349 1382 1563 1671

32 122 239 316 350 415 695 792 823 945 1175 1215 1340 1426 1551 1684

38 158 202 309 334 455 715 802 826 919 1163 1256 1309 1416 1564 1655

49 97 198 271 400 459 702 808 846 911 1176 1229 1330 1411 1615 1630

6 106 222 278 397 474 728 739 833 893 1147 1264 1366 1402 1590 1644

46 133 196 280 358 411 678 776 859 938 1202 1253 1305 1451 1604 1625

50 110 188 258 402 470 666 736 889 918 1136 1278 1314 1392 1585 1623

65 115 170 301 392 440 679 740 877 950 1197 1263 1341 1428 1583 1656

2 153 215 297 387 469 650 755 890 960 1195 1235 1318 1447 1616 1690

61 96 195 321 378 485 705 772 861 966 1148 1266 1323 1456 1570 1628

31 95 227 295 347 461 719 751 836 897 1182 1288 1361 1403 1588 1621

60 89 237 287 368 473 700 801 850 934 1200 1272 1304 1380 1581 1663

76 128 163 269 404 439 698 750 831 894 1193 1258 1303 1385 1543 1670

52 136 174 314 343 467 651 766 829 898 1155 1255 1297 1423 1550 1672

64 119 211 294 352 405 685 742 862 913 1162 1216 1336 1446 1552 1650

30 86 171 246 379 428 703 754 816 930 1164 1260 1344 1445 1610 1693

58 134 175 256 356 481 696 800 834 909 1142 1250 1327 1439 1573 1689

0 137 190 262 361 436 658 748 827 959 1185 1245 1374 1398 1569 1633

19 157 207 273 399 425 665 729 869 908 1181 1236 1342 1406 1593 1687

72 144 186 310 342 450 667 789 876 924 1205 1285 1345 1389 1567 1679

27 90 236 270 341 435 725 762 878 900 1179 1226 1365 1436 1559 1661

16 120 185 274 335 407 688 797 856 912 1171 1262 1352 1404 1541 1626

41 108 201 289 374 438 684 786 819 958 1153 1281 1298 1407 1586 1686

26 121 177 306 382 460 708 731 815 906 1198 1290 1328 1427 1566 1638

78 92 189 285 365 444 682 796 839 891 1178 1237 1316 1414 1598 1648

29 147 235 255 332 430 674 768 813 947 1210 1294 1329 1440 1608 1654

51 161 183 284 380 427 656 799 885 920 1140 1219 1300 1390 1614 1665

35 142 162 300 383 468 701 741 867 955 1146 1257 1355 1378 1545 1622

21 140 224 276 403 432 681 805 832 944 1157 1280 1369 1391 1582 1662

18 93 197 288 390 422 713 791 812 969 1194 1279 1350 1442 1542 1666

59 127 232 254 336 417 723 788 844 954 1154 1273 1348 1417 1546 1681

23 145 221 282 366 408 649 749 854 926 1158 1232 1301 1431 1561 1698

13 138 166 243 354 457 660 793 860 957 1173 1240 1335 1412 1578 1677

8 100 231 323 367 478 697 783 871 899 1190 1223 1353 1410 1557 1647

79 107 203 296 338 434 657 778 828 963 1169 1270 1346 1443 1607 1676

48 109 234 251 393 453 661 769 868 949 1139 1238 1308 1397 1556 1692

69 87 176 320 327 462 676 738 872 946 1168 1241 1315 1415 1572 1668

25 130 240 265 388 409 693 759 887 907 1184 1261 1317 1408 1548 1680

44 126 226 250 386 466 672 795 824 951 1160 1248 1375 1450 1560 1646

53 150 223 302 339 471 722 734 883 941 1172 1274 1338 1457 1606 1674

80 124 184 253 373 429 671 743 853 936 1138 1224 1334 1379 1554 1620

57 116 228 275 391 452 687 770 882 927 1166 1292 1358 1437 1539 1635

62 98 218 259 384 451 663 747 818 896 1134 1225 1332 1400 1595 1688

20 143 213 245 346 445 675 752 874 917 1207 1276 1324 1396 1568 1643

43 123 204 322 353 484 721 790 886 953 1180 1251 1306 1420 1603 1632

42 155 173 283 355 412 669 733 852 892 1135 1265 1351 1394 1592 1657

36 85 194 247 333 475 648 732 880 901 1204 1246 1331 1386 1617 1642

75 91 230 317 376 442 710 806 810 928 1149 1244 1363 1448 1602 1694

3 102 169 312 372 410 683 765 841 905 1214 1277 1373 1413 1574 1645

66 139 178 303 396 413 718 773 814 910 1186 1231 1299 1393 1605 1667

33 99 205 272 370 433 707 756 849 948 1137 1249 1310 1425 1547 1651

1 103 182 293 362 420 652 803 838 971 1159 1242 1347 1435 1611 1637

4 118 187 249 344 446 717 771 863 970 1143 1284 1307 1441 1597 1634

24 135 225 268 389 476 689 774 848 964 1209 1291 1311 1452 1594 1675

11 114 168 277 369 464 720 794 820 923 1206 1293 1326 1409 1555 1639

37 84 167 304 401 477 662 781 851 931 1167 1271 1343 1449 1599 1629

16 149 221 320 395 425 502 786 838 944 977 1253 1357 1419 1632 1779

38 85 168 305 402 478 553 782 852 932 1022 1272 1344 1450 1630 1709

68 114 242 282 332 449 520 808 874 915 1038 1235 1323 1454 1625 1728

56 130 201 287 338 424 488 758 858 962 1014 1231 1373 1389 1696 1737

69 106 209 245 349 438 491 746 844 930 1026 1255 1322 1406 1665 1764

40 118 192 268 386 419 511 759 841 933 992 1229 1338 1385 1686 1741

15 84 239 267 346 417 498 730 882 953 1020 1221 1314 1435 1642 1746

29 112 207 261 350 450 524 785 846 940 972 1283 1326 1384 1661 1704

10 81 210 300 365 484 554 799 836 966 981 1248 1372 1400 1670 1727

8 153 230 308 382 422 542 780 831 916 1034 1228 1320 1456 1697 1726

41 126 217 291 361 415 555 778 822 904 989 1260 1296 1388 1674 1720

75 161 163 25 833 1457 526 731 871 917 978 1270 1361 1434 1679 1759

13 150 193 306 360 464 501 765 812 968 1003 1276 1334 1382 1637 1767

6 95 181 309 376 466 515 783 848 943 988 1287 1369 1377 1660 1750

48 160 194 249 352 444 496 776 867 957 1040 1244 1358 1423 1659 1717

55 132 165 316 364 407 494 738 876 938 991 1284 1303 1396 1653 1765

57 83 220 262 330 483 527 745 823 936 1013 1288 1368 1431 1692 1768

35 105 234 292 358 427 561 747 880 969 997 1223 1340 1420 1700 1708

78 89 215 280 324 481 499 805 885 923 983 1240 1371 1445 1683 1775

74 155 213 293 399 473 492 768 843 941 980 1219 1313 1430 1650 1721

18 152 166 264 372 455 534 764 866 934 1021 1269 1297 1402 1698 1751

72 113 200 319 327 420 541 788 865 896 985 1218 1363 1433 1621 1739

64 157 218 253 396 480 543 762 859 903 975 1234 1360 1455 1641 1752

46 147 211 314 341 432 521 754 818 905 1050 1290 1321 1439 1628 1723

11 142 173 312 326 442 564 736 826 963 1041 1222 1365 1425 1654 1778

71 133 180 265 378 448 560 781 889 926 1010 1268 1355 1422 1684 1712

23 102 182 299 329 459 504 761 856 922 1031 1216 1350 1383 1672 1773

33 123 240 317 351 416 558 793 824 946 987 1215 1341 1427 1685 1771

39 159 203 310 335 456 550 803 827 920 1006 1257 1310 1417 1656 1724

50 98 199 272 401 460 532 809 847 912 1015 1230 1331 1412 1631 1758

7 107 223 279 398 475 497 740 834 894 1042 1265 1367 1403 1645 1776

47 134 197 281 359 412 557 777 860 939 1019 1254 1306 1452 1626 1769

51 111 189 259 403 471 509 737 890 919 1024 1279 1315 1393 1624 1731

66 116 171 302 393 441 519 741 878 951 982 1264 1342 1429 1657 1738

3 154 216 298 388 470 525 756 811 961 1005 1236 1319 1448 1691 1740

62 97 196 322 379 406 536 773 862 967 1004 1267 1324 1457 1629 1718

32 96 228 296 348 462 493 752 837 898 998 1289 1362 1404 1622 1761

61 90 238 288 369 474 533 802 851 935 1037 1273 1305 1381 1664 1757

77 129 164 270 325 440 537 751 832 895 1045 1259 1304 1386 1671 1781

53 137 175 315 344 468 552 767 830 899 1028 1256 1298 1424 1673 1755

65 120 212 295 353 405 489 743 863 914 995 1217 1337 1447 1651 1747

31 87 172 247 380 429 548 755 817 931 1043 1261 1345 1446 1694 1729

59 135 176 257 357 482 518 801 835 910 1046 1251 1328 1440 1690 1774

0 138 191 263 362 437 547 749 828 960 986 1246 1375 1399 1634 1754

20 158 208 274 400 426 563 729 870 909 973 1237 1343 1407 1688 1706

73 145 187 311 343 451 539 790 877 925 999 1286 1346 1390 1680 1716

28 91 237 271 342 436 551 763 879 901 1029 1227 1366 1437 1662 1722

17 121 186 275 336 408 517 798 857 913 1017 1263 1353 1405 1627 1733

42 109 202 290 375 439 545 787 820 959 1030 1282 1299 1408 1687 1770

27 122 178 307 383 461 486 732 816 907 1001 1291 1329 1428 1639 1730

79 93 190 286 366 445 506 797 840 891 976 1238 1317 1415 1649 1734

30 148 236 256 333 431 559 769 814 948 990 1295 1330 1441 1655 1749

52 82 184 285 381 428 514 800 886 921 1051 1220 1301 1391 1666 1766

36 143 162 301 384 469 503 742 868 956 1049 1258 1356 1379 1623 1745

22 141 225 277 404 433 528 806 833 945 1002 1281 1370 1392 1663 1715

19 94 198 289 391 423 513 792 813 970 1036 1280 1351 1443 1667 1744

60 128 233 255 337 418 565 789 845 955 974 1274 1349 1418 1682 1760

24 146 222 283 367 409 516 750 855 927 1047 1233 1302 1432 1699 1736

14 139 167 243 355 458 538 794 861 958 1009 1241 1336 1413 1678 1748

9 101 232 244 368 479 522 784 872 900 1016 1224 1354 1411 1648 1714

80 108 204 297 339 435 508 779 829 964 1018 1271 1347 1444 1677 1742

49 110 235 252 394 454 505 770 869 950 996 1239 1309 1398 1693 1701

70 88 177 321 328 463 546 739 873 947 1039 1242 1316 1416 1669 1703

26 131 241 266 389 410 510 760 888 908 1035 1262 1318 1409 1681 1756

45 127 227 251 387 467 500 796 825 952 979 1249 1376 1451 1647 1711

54 151 224 303 340 472 495 735 884 942 1033 1275 1339 1378 1675 1780

1 125 185 254 374 430 566 744 854 937 1025 1225 1335 1380 1620 1725

58 117 229 276 392 453 535 771 883 928 1007 1293 1359 1438 1636 1710

63 99 219 260 385 452 556 748 819 897 1052 1226 1333 1401 1689 1762

21 144 214 246 347 446 512 753 875 918 1032 1277 1325 1397 1644 1713

44 124 205 323 354 485 531 791 887 954 984 1252 1307 1421 1633 1735

43 156 174 284 356 413 540 734 853 893 994 1266 1352 1395 1658 1719

37 86 195 248 334 476 487 733 881 902 1000 1247 1332 1387 1643 1705

76 92 231 318 377 443 544 807 810 929 1011 1245 1364 1449 1695 1702

4 103 170 313 373 411 549 766 842 906 1048 1278 1374 1414 1646 1743

67 140 179 304 397 414 507 774 815 911 1008 1232 1300 1394 1668 1707

34 100 206 273 371 434 530 757 850 949 1012 1250 1311 1426 1652 1777

2 104 183 294 363 421 529 804 839 892 1027 1243 1348 1436 1638 1772

5 119 188 250 345 447 523 772 864 971 1044 1285 1308 1442 1635 1763

25 136 226 269 390 477 562 775 849 965 1023 1292 1312 1453 1676 1732

12 115 169 278 370 465 490 795 821 924 993 1294 1327 1410 1640 1753

17 150 222 321 396 426 503 618 839 945 978 1065 1358 1420 1780 1786

13 116 170 279 371 466 491 628 822 925 994 1070 1328 1411 1754 1858

39 86 169 306 403 479 554 647 853 933 1023 1093 1345 1451 1710 1811

69 115 163 283 333 450 521 576 875 916 1039 1092 1324 1455 1729 1845

57 131 202 288 339 425 489 603 859 963 1015 1086 1374 1390 1738 1783

70 107 210 246 350 439 492 580 845 931 1027 1125 1323 1407 1765 1856

41 119 193 269 387 420 512 585 842 934 993 1133 1339 1386 1742 1818

16 85 240 268 347 418 499 623 883 954 1021 1116 1315 1436 1747 1825

30 113 208 262 351 451 525 646 847 941 972 1083 1327 1385 1705 1827

11 81 211 301 366 485 555 645 837 967 982 1131 1373 1401 1728 1805

9 154 231 309 383 423 543 639 832 917 1035 1054 1321 1457 1727 1848

42 127 218 292 362 416 556 598 823 905 990 1074 1296 1389 1721 1844

76 82 164 259 332 458 527 606 872 918 979 1061 1362 1435 1760 1788

14 151 194 307 361 465 502 589 813 969 1004 1087 1335 1383 1768 1842

7 96 182 310 377 467 516 636 849 944 989 1117 1370 1377 1751 1834

49 161 195 250 353 445 497 604 868 958 1041 1105 1359 1424 1718 1816

56 133 166 317 365 408 495 607 877 939 992 1118 1304 1397 1766 1861

58 84 221 263 331 484 528 627 824 937 1014 1089 1369 1432 1769 1841

36 106 235 293 359 428 562 614 881 970 998 1064 1341 1421 1709 1793

79 90 216 281 324 482 500 640 886 924 984 1078 1372 1446 1776 1803

75 156 214 294 400 474 493 590 844 942 981 1059 1314 1431 1722 1809

19 153 167 265 373 456 535 578 867 935 1022 1057 1298 1403 1752 1820

73 114 201 320 328 421 542 591 866 897 986 1090 1364 1434 1740 1857

65 158 219 254 397 481 544 642 860 904 976 1124 1361 1456 1753 1817

47 148 212 315 342 433 522 617 819 906 1051 1062 1322 1440 1724 1821

12 143 174 313 327 443 565 631 827 964 1042 1055 1366 1426 1779 1836

72 134 181 266 379 449 561 612 890 927 1011 1097 1356 1423 1713 1853

24 103 183 300 330 460 505 610 857 923 1032 1104 1351 1384 1774 1832

34 124 241 318 352 417 559 643 825 947 988 1106 1342 1428 1772 1802

40 160 204 311 336 457 551 597 828 921 1007 1084 1311 1418 1725 1831

51 99 200 273 402 461 533 615 848 913 1016 1127 1332 1413 1759 1847

8 108 224 280 399 476 498 608 835 895 1043 1123 1368 1404 1777 1823

48 135 198 282 360 413 558 570 861 940 1020 1067 1307 1453 1770 1835

52 112 190 260 404 472 510 577 811 920 1025 1121 1316 1394 1732 1801

67 117 172 303 394 442 520 579 879 952 983 1113 1343 1430 1739 1829

4 155 217 299 389 471 526 637 812 962 1006 1095 1320 1449 1741 1782

63 98 197 323 380 407 537 600 863 968 1005 1060 1325 1378 1719 1790

33 97 229 297 349 463 494 596 838 899 999 1120 1363 1405 1762 1843

62 91 239 289 370 475 534 620 852 936 1038 1072 1306 1382 1758 1798

78 130 165 271 326 441 538 594 833 896 1046 1082 1305 1387 1702 1787

54 138 176 316 345 469 553 586 831 900 1029 1088 1299 1425 1756 1812

66 121 213 296 354 405 490 568 864 915 996 1099 1338 1448 1748 1797

32 88 173 248 381 430 549 613 818 932 1044 1056 1346 1447 1730 1849

60 136 177 258 358 483 519 593 836 911 1047 1096 1329 1441 1775 1800

0 139 192 264 363 438 548 625 829 961 987 1100 1376 1400 1755 1822

21 159 209 275 401 427 564 635 871 910 974 1115 1344 1408 1707 1806

74 146 188 312 344 452 540 641 878 926 1000 1132 1347 1391 1717 1792

29 92 238 272 343 437 552 572 880 902 1030 1111 1367 1438 1723 1789

18 122 187 276 337 409 518 609 858 914 1018 1081 1354 1406 1734 1830

43 110 203 291 376 440 546 569 821 960 1031 1110 1300 1409 1771 1794

28 123 179 308 384 462 486 573 817 908 1002 1126 1330 1429 1731 1784

80 94 191 287 367 446 507 588 841 891 977 1102 1318 1416 1735 1859

31 149 237 257 334 432 560 605 815 949 991 1114 1331 1442 1750 1850

53 83 185 286 382 429 515 584 887 922 1052 1080 1302 1392 1767 1819

37 144 162 302 385 470 504 634 869 957 1050 1108 1357 1380 1746 1840

23 142 226 278 325 434 529 583 834 946 1003 1053 1371 1393 1716 1796

20 95 199 290 392 424 514 599 814 971 1037 1069 1352 1444 1745 1815

61 129 234 256 338 419 566 575 846 956 975 1122 1350 1419 1761 1824

25 147 223 284 368 410 517 587 856 928 1048 1077 1303 1433 1737 1851

15 140 168 243 356 459 539 633 862 959 1010 1066 1337 1414 1749 1828

10 102 233 245 369 480 523 581 873 901 1017 1091 1355 1412 1715 1833

1 109 205 298 340 436 509 567 830 965 1019 1076 1348 1445 1743 1791

50 111 236 253 395 455 506 622 870 951 997 1128 1310 1399 1701 1814

71 89 178 322 329 464 547 595 874 948 1040 1079 1317 1417 1704 1813

27 132 242 267 390 411 511 630 889 909 1036 1101 1319 1410 1757 1807

46 128 228 252 388 468 501 619 826 953 980 1085 1297 1452 1712 1846

55 152 225 304 341 473 496 644 885 943 1034 1071 1340 1379 1781 1854

2 126 186 255 375 431 487 629 855 938 1026 1068 1336 1381 1726 1837

59 118 230 277 393 454 536 601 884 929 1008 1109 1360 1439 1711 1804

64 100 220 261 386 453 557 632 820 898 973 1073 1334 1402 1763 1852

22 145 215 247 348 447 513 574 876 919 1033 1063 1326 1398 1714 1855

45 125 206 244 355 406 532 638 888 955 985 1058 1308 1422 1736 1795

44 157 175 285 357 414 541 624 854 894 995 1129 1353 1396 1720 1862

38 87 196 249 335 477 488 621 882 903 1001 1098 1333 1388 1706 1808

77 93 232 319 378 444 545 582 810 930 1012 1119 1365 1450 1703 1838

5 104 171 314 374 412 550 626 843 907 1049 1075 1375 1415 1744 1826

68 141 180 305 398 415 508 616 816 912 1009 1094 1301 1395 1708 1839

35 101 207 274 372 435 531 611 851 950 1013 1103 1312 1427 1778 1810

3 105 184 295 364 422 530 602 840 893 1028 1130 1349 1437 1773 1785

6 120 189 251 346 448 524 571 865 892 1045 1107 1309 1443 1764 1799

26 137 227 270 391 478 563 592 850 966 1024 1112 1313 1454 1733 1860

18 151 223 322 397 427 504 619 709 946 979 1066 1206 1421 1787 1863

27 138 228 271 392 479 564 593 692 967 1025 1113 1212 1455 1861 1865

14 117 171 280 372 467 492 629 723 926 995 1071 1209 1412 1859 1866

40 87 170 307 404 480 555 568 665 934 1024 1094 1170 1452 1812 1867

70 116 164 284 334 451 522 577 649 917 1040 1093 1214 1456 1846 1868

58 132 203 289 340 426 490 604 715 964 1016 1087 1204 1391 1784 1869

71 108 211 247 351 440 493 581 712 932 1028 1126 1199 1408 1857 1870

42 120 194 270 388 421 513 586 673 935 994 1054 1190 1387 1819 1871

17 86 241 269 348 419 500 624 717 955 1022 1117 1159 1437 1826 1872

31 114 209 263 352 452 526 647 707 942 972 1084 1180 1386 1828 1873

12 81 212 302 367 406 556 646 702 968 983 1132 1136 1402 1806 1874

10 155 232 310 384 424 544 640 693 918 1036 1055 1155 1378 1849 1875

43 128 219 293 363 417 557 599 662 906 991 1075 1164 1390 1845 1876

77 83 165 260 333 459 528 607 683 919 980 1062 1191 1436 1789 1877

15 152 195 308 362 466 503 590 719 970 1005 1088 1168 1384 1843 1878

8 97 183 311 378 468 517 637 658 945 990 1118 1173 1377 1835 1879

50 82 196 251 354 446 498 605 667 959 1042 1106 1211 1425 1817 1880

57 134 167 318 366 409 496 608 694 940 993 1119 1154 1398 1862 1881

59 85 222 264 332 485 529 628 671 938 1015 1090 1153 1433 1842 1882

37 107 236 294 360 429 563 615 676 971 999 1065 1147 1422 1794 1883

80 91 217 282 324 483 501 641 714 925 985 1079 1186 1447 1804 1884

76 157 215 295 401 475 494 591 657 943 982 1060 1194 1432 1810 1885

20 154 168 266 374 457 536 579 656 936 1023 1058 1177 1404 1821 1886

74 115 202 321 329 422 543 592 650 898 987 1091 1144 1435 1858 1887

66 159 220 255 398 482 545 643 689 905 977 1125 1192 1457 1818 1888

48 149 213 316 343 434 523 618 697 907 1052 1063 1195 1441 1822 1889

13 144 175 314 328 444 566 632 680 965 1043 1056 1135 1427 1837 1890

73 135 182 267 380 450 562 613 727 928 1012 1098 1202 1424 1854 1891

25 104 184 301 331 461 506 611 695 924 1033 1105 1148 1385 1833 1892

35 125 242 319 353 418 560 644 698 948 989 1107 1178 1429 1803 1893

41 161 205 312 337 458 552 598 718 922 1008 1085 1166 1419 1832 1894

52 100 201 274 403 462 534 616 705 914 1017 1128 1179 1414 1848 1895

9 109 225 281 400 477 499 609 651 896 1044 1124 1150 1405 1824 1896

49 136 199 283 361 414 559 571 681 941 1021 1068 1205 1454 1836 1897

53 113 191 261 325 473 511 578 669 921 1026 1122 1139 1395 1802 1898

68 118 173 304 395 443 521 580 682 953 984 1114 1200 1431 1830 1899

5 156 218 300 390 472 527 638 653 963 1007 1096 1198 1450 1782 1900

64 99 198 244 381 408 538 601 708 969 1006 1061 1151 1379 1791 1901

34 98 230 298 350 464 495 597 722 900 1000 1121 1185 1406 1844 1902

63 92 240 290 371 476 535 621 703 937 1039 1073 1203 1383 1799 1903

79 131 166 272 327 442 539 595 701 897 1047 1083 1196 1388 1788 1904

55 139 177 317 346 470 554 587 654 901 1030 1089 1158 1426 1813 1905

67 122 214 297 355 405 491 569 688 916 997 1100 1165 1449 1798 1906

33 89 174 249 382 431 550 614 706 933 1045 1057 1167 1448 1850 1907

61 137 178 259 359 484 520 594 699 912 1048 1097 1145 1442 1801 1908

0 140 193 265 364 439 549 626 661 962 988 1101 1188 1401 1823 1909

22 160 210 276 402 428 565 636 668 911 975 1116 1184 1409 1807 1910

75 147 189 313 345 453 541 642 670 927 1001 1133 1208 1392 1793 1911

30 93 239 273 344 438 553 573 728 903 1031 1112 1182 1439 1790 1912

19 123 188 277 338 410 519 610 691 915 1019 1082 1174 1407 1831 1913

44 111 204 292 377 441 547 570 687 961 1032 1111 1156 1410 1795 1914

29 124 180 309 385 463 486 574 711 909 1003 1127 1201 1430 1785 1915

1 95 192 288 368 447 508 589 685 891 978 1103 1181 1417 1860 1916

32 150 238 258 335 433 561 606 677 950 992 1115 1213 1443 1851 1917

54 84 186 287 383 430 516 585 659 923 973 1081 1143 1393 1820 1918

38 145 162 303 386 471 505 635 704 958 1051 1109 1149 1381 1841 1919

24 143 227 279 326 435 530 584 684 947 1004 1053 1160 1394 1797 1920

21 96 200 291 393 425 515 600 716 892 1038 1070 1197 1445 1816 1921

62 130 235 257 339 420 487 576 726 957 976 1123 1157 1420 1825 1922

26 148 224 285 369 411 518 588 652 929 1049 1078 1161 1434 1852 1923

16 141 169 243 357 460 540 634 663 960 1011 1067 1176 1415 1829 1924

11 103 234 246 370 481 524 582 700 902 1018 1092 1193 1413 1834 1925

2 110 206 299 341 437 510 567 660 966 1020 1077 1172 1446 1792 1926

51 112 237 254 396 456 507 623 664 952 998 1129 1142 1400 1815 1927

72 90 179 323 330 465 548 596 679 949 1041 1080 1171 1418 1814 1928

28 133 163 268 391 412 512 631 696 910 1037 1102 1187 1411 1808 1929

47 129 229 253 389 469 502 620 675 954 981 1086 1163 1453 1847 1930

56 153 226 305 342 474 497 645 725 944 1035 1072 1175 1380 1855 1931

3 127 187 256 376 432 488 630 674 939 1027 1069 1141 1382 1838 1932

60 119 231 278 394 455 537 602 690 930 1009 1110 1169 1440 1805 1933

65 101 221 262 387 454 558 633 666 899 974 1074 1134 1403 1853 1934

23 146 216 248 349 448 514 575 678 920 1034 1064 1210 1399 1856 1935

46 126 207 245 356 407 533 639 724 956 986 1059 1183 1423 1796 1936

45 158 176 286 358 415 542 625 672 895 996 1130 1138 1397 1783 1937

39 88 197 250 336 478 489 622 648 904 1002 1099 1207 1389 1809 1938

78 94 233 320 379 445 546 583 713 931 1013 1120 1152 1451 1839 1939

6 105 172 315 375 413 551 627 686 908 1050 1076 1137 1416 1827 1940

69 142 181 306 399 416 509 617 721 913 1010 1095 1189 1396 1840 1941

36 102 208 275 373 436 532 612 710 951 1014 1104 1140 1428 1811 1942

4 106 185 296 365 423 531 603 655 894 1029 1131 1162 1438 1786 1943

7 121 190 252 347 449 525 572 720 893 1046 1108 1146 1444 1800 1864

19 152 224 323 398 428 505 620 710 789 980 1067 1207 1256 1863

8 122 191 253 348 450 526 573 721 775 1047 1109 1147 1288 1865

28 139 229 272 393 480 565 594 693 778 1026 1114 1213 1295 1866

15 118 172 281 373 468 493 630 724 798 996 1072 1210 1217 1867

41 88 171 308 325 481 556 569 666 785 1025 1095 1171 1275 1868

71 117 165 285 335 452 523 578 650 731 1041 1094 1135 1238 1869

59 133 204 290 341 427 491 605 716 761 1017 1088 1205 1234 1870

72 109 212 248 352 441 494 582 713 749 1029 1127 1200 1258 1871

43 121 195 271 389 422 514 587 674 762 995 1055 1191 1232 1872

18 87 242 270 349 420 501 625 718 733 1023 1118 1160 1224 1873

32 115 210 264 353 453 527 568 708 788 972 1085 1181 1286 1874

13 81 213 303 368 407 557 647 703 802 984 1133 1137 1251 1875

11 156 233 311 385 425 545 641 694 783 1037 1056 1156 1231 1876

44 129 220 294 364 418 558 600 663 781 992 1076 1165 1263 1877

78 84 166 261 334 460 529 608 684 734 981 1063 1192 1273 1878

16 153 196 309 363 467 504 591 720 768 1006 1089 1169 1279 1879

9 98 184 312 379 469 518 638 659 786 991 1119 1174 1290 1880

51 83 197 252 355 447 499 606 668 779 1043 1107 1212 1247 1881

58 135 168 319 367 410 497 609 695 741 994 1120 1155 1287 1882

60 86 223 265 333 406 530 629 672 748 1016 1091 1154 1291 1883

38 108 237 295 361 430 564 616 677 750 1000 1066 1148 1226 1884

1 92 218 283 324 484 502 642 715 808 986 1080 1187 1243 1885

77 158 216 296 402 476 495 592 658 771 983 1061 1195 1222 1886

21 155 169 267 375 458 537 580 657 767 1024 1059 1178 1272 1887

75 116 203 322 330 423 544 593 651 791 988 1092 1145 1221 1888

67 160 221 256 399 483 546 644 690 765 978 1126 1193 1237 1889

49 150 214 317 344 435 524 619 698 757 973 1064 1196 1293 1890

14 145 176 315 329 445 487 633 681 739 1044 1057 1136 1225 1891

74 136 183 268 381 451 563 614 728 784 1013 1099 1203 1271 1892

26 105 185 302 332 462 507 612 696 764 1034 1106 1149 1219 1893

36 126 163 320 354 419 561 645 699 796 990 1108 1179 1215 1894

42 82 206 313 338 459 553 599 719 806 1009 1086 1167 1260 1895

53 101 202 275 404 463 535 617 706 732 1018 1129 1180 1233 1896

10 110 226 282 401 478 500 610 652 743 1045 1125 1151 1268 1897

50 137 200 284 362 415 560 572 682 780 1022 1069 1206 1257 1898

54 114 192 262 326 474 512 579 670 740 1027 1123 1140 1282 1899

69 119 174 305 396 444 522 581 683 744 985 1115 1201 1267 1900

6 157 219 301 391 473 528 639 654 759 1008 1097 1199 1239 1901

65 100 199 245 382 409 539 602 709 776 1007 1062 1152 1270 1902

35 99 231 299 351 465 496 598 723 755 1001 1122 1186 1292 1903

64 93 241 291 372 477 536 622 704 805 1040 1074 1204 1276 1904

80 132 167 273 328 443 540 596 702 754 1048 1084 1197 1262 1905

56 140 178 318 347 471 555 588 655 770 1031 1090 1159 1259 1906

68 123 215 298 356 405 492 570 689 746 998 1101 1166 1220 1907

34 90 175 250 383 432 551 615 707 758 1046 1058 1168 1264 1908

62 138 179 260 360 485 521 595 700 804 1049 1098 1146 1254 1909

0 141 194 266 365 440 550 627 662 752 989 1102 1189 1249 1910

23 161 211 277 403 429 566 637 669 729 976 1117 1185 1240 1911

76 148 190 314 346 454 542 643 671 793 1002 1054 1209 1289 1912

31 94 240 274 345 439 554 574 649 766 1032 1113 1183 1230 1913

20 124 189 278 339 411 520 611 692 801 1020 1083 1175 1266 1914

45 112 205 293 378 442 548 571 688 790 1033 1112 1157 1285 1915

30 125 181 310 386 464 486 575 712 735 1004 1128 1202 1294 1916

2 96 193 289 369 448 509 590 686 800 979 1104 1182 1241 1917

33 151 239 259 336 434 562 607 678 772 993 1116 1214 1218 1918

55 85 187 288 384 431 517 586 660 803 974 1082 1144 1223 1919

39 146 162 304 387 472 506 636 705 745 1052 1110 1150 1261 1920

25 144 228 280 327 436 531 585 685 809 1005 1053 1161 1284 1921

22 97 201 292 394 426 516 601 717 795 1039 1071 1198 1283 1922

63 131 236 258 340 421 488 577 727 792 977 1124 1158 1277 1923

27 149 225 286 370 412 519 589 653 753 1050 1079 1162 1236 1924

17 142 170 243 358 461 541 635 664 797 1012 1068 1177 1244 1925

12 104 235 247 371 482 525 583 701 787 1019 1093 1194 1227 1926

3 111 207 300 342 438 511 567 661 782 1021 1078 1173 1274 1927

52 113 238 255 397 457 508 624 665 773 999 1130 1143 1242 1928

73 91 180 244 331 466 549 597 680 742 1042 1081 1172 1245 1929

29 134 164 269 392 413 513 632 697 763 1038 1103 1188 1265 1930

48 130 230 254 390 470 503 621 676 799 982 1087 1164 1252 1931

57 154 227 306 343 475 498 646 726 738 1036 1073 1176 1278 1932

4 128 188 257 377 433 489 631 675 747 1028 1070 1142 1228 1933

61 120 232 279 395 456 538 603 691 774 1010 1111 1170 1216 1934

66 102 222 263 388 455 559 634 667 751 975 1075 1134 1229 1935

24 147 217 249 350 449 515 576 679 756 1035 1065 1211 1280 1936

47 127 208 246 357 408 534 640 725 794 987 1060 1184 1255 1937

46 159 177 287 359 416 543 626 673 737 997 1131 1139 1269 1938

40 89 198 251 337 479 490 623 648 736 1003 1100 1208 1250 1939

79 95 234 321 380 446 547 584 714 730 1014 1121 1153 1248 1940

7 106 173 316 376 414 552 628 687 769 1051 1077 1138 1281 1941

70 143 182 307 400 417 510 618 722 777 1011 1096 1190 1235 1942

37 103 209 276 374 437 533 613 711 760 1015 1105 1141 1253 1943

5 107 186 297 366 424 532 604 656 807 1030 1132 1163 1246 1864

7. based on GRS abnormal LDPC (1944,1620) sign indicating number (7), R=5/6

21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21

21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21

21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22

0 81 162 243 324 405 486 567 648 729 810 891 972 1053 1134 1215 1296 1377 1458 15391620

14 88 239 245 376 462 526 640 671 796 816 920 1052 1123 1160 1269 1331 1427 1487 15431654

15 89 240 246 377 463 527 641 672 797 817 921 973 1124 1161 1270 1332 1428 1488 15441655

16 90 241 247 378 464 528 642 673 798 818 922 974 1125 1162 1271 1333 1429 1489 15451656

17 91 242 248 379 465 529 643 674 799 819 923 975 1126 1163 1272 1334 1430 1490 15461657

18 92 163 249 380 466 530 644 675 800 820 924 976 1127 1164 1273 1335 1431 1491 15471658

19 93 164 250 381 467 531 645 676 801 821 925 977 1128 1165 1274 1336 1432 1492 15481659

20 94 165 251 382 468 532 646 677 802 822 926 978 1129 1166 1275 1337 1433 1493 15491660

21 95 166 252 383 469 533 647 678 803 823 927 979 1130 1167 1276 1338 1434 1494 15501661

22 96 167 253 384 470 534 568 679 804 824 928 980 1131 1168 1277 1339 1435 1495 15511662

23 97 168 254 385 471 535 569 680 805 825 929 981 1132 1169 1278 1340 1436 1496 15521663

24 98 169 255 386 472 536 570 681 806 826 930 982 1133 1170 1279 1341 1437 1497 15531664

25 99 170 256 387 473 537 571 682 807 827 931 983 1054 1171 1280 1342 1438 1498 15541665

26 100 171 257 388 474 538 572 683 808 828 932 984 1055 1172 1281 1343 1439 1499 15551666

27 101 172 258 389 475 539 573 684 809 829 933 985 1056 1173 1282 1344 1440 1500 15561667

28 102 173 259 390 476 540 574 685 730 830 934 986 1057 1174 1283 1345 1441 1501 15571668

29 103 174 260 391 477 541 575 686 731 831 935 987 1058 1175 1284 1346 1442 1502 15581669

30 104 175 261 392 478 542 576 687 732 832 936 988 1059 1176 1285 1347 1443 1503 15591670

31 105 176 262 393 479 543 577 688 733 833 937 989 1060 1177 1286 1348 1444 1504 15601671

32 106 177 263 394 480 544 578 689 734 834 938 990 1061 1178 1287 1349 1445 1505 15611672

33 107 178 264 395 481 545 579 690 735 835 939 991 1062 1179 1288 1350 1446 1506 15621673

34 108 179 265 396 482 546 580 691 736 836 940 992 1063 1180 1289 1351 1447 1507 15631674

35 109 180 266 397 483 547 581 692 737 837 941 993 1064 1181 1290 1352 1448 1508 15641675

36 110 181 267 398 484 548 582 693 738 838 942 994 1065 1182 1291 1353 1449 1509 15651676

37 111 182 268 399 485 549 583 694 739 839 943 995 1066 1183 1292 1354 1450 1510 15661677

38 112 183 269 400 406 550 584 695 740 840 944 996 1067 1184 1293 1355 1451 1511 15671678

39 113 184 270 401 407 551 585 696 741 841 945 997 1068 1185 1294 1356 1452 1512 15681679

40 114 185 271 402 408 552 586 697 742 842 946 998 1069 1186 1295 1357 1453 1513 15691680

41 115 186 272 403 409 553 587 698 743 843 947 999 1070 1187 1216 1358 1454 1514 15701681

42 116 187 273 404 410 554 588 699 744 844 948 1000 1071 1188 1217 1359 1455 1515 15711682

43 117 188 274 325 411 555 589 700 745 845 949 1001 1072 1189 1218 1360 1456 1516 15721683

44 118 189 275 326 412 556 590 701 746 846 950 1002 1073 1190 1219 1361 1457 1517 15731684

45 119 190 276 327 413 557 591 702 747 847 951 1003 1074 1191 1220 1362 1378 1518 15741685

46 120 191 277 328 414 558 592 703 748 848 952 1004 1075 1192 1221 1363 1379 1519 15751686

47 121 192 278 329 415 559 593 704 749 849 953 1005 1076 1193 1222 1364 1380 1520 15761687

48 122 193 279 330 416 560 594 705 750 850 954 1006 1077 1194 1223 1365 1381 1521 15771688

49 123 194 280 331 417 561 595 706 751 851 955 1007 1078 1195 1224 1366 1382 1522 15781689

50 124 195 281 332 418 562 596 707 752 852 956 1008 1079 1196 1225 1367 1383 1523 15791690

51 125 196 282 333 419 563 597 708 753 853 957 1009 1080 1197 1226 1368 1384 1524 15801691

52 126 197 283 334 420 564 598 709 754 854 958 1010 1081 1198 1227 1369 1385 1525 15811692

53 127 198 284 335 421 565 599 710 755 855 959 1011 1082 1199 1228 1370 1386 1526 15821693

54 128 199 285 336 422 566 600 711 756 856 960 1012 1083 1200 1229 1371 1387 1527 15831694

55 129 200 286 337 423 487 601 712 757 857 961 1013 1084 1201 1230 1372 1388 1528 15841695

56 130 201 287 338 424 488 602 713 758 858 962 1014 1085 1202 1231 1373 1389 1529 15851696

57 131 202 288 339 425 489 603 714 759 859 963 1015 1086 1203 1232 1374 1390 1530 15861697

58 132 203 289 340 426 490 604 715 760 860 964 1016 1087 1204 1233 1375 1391 1531 15871698

59 133 204 290 341 427 491 605 716 761 861 965 1017 1088 1205 1234 1376 1392 1532 15881699

60 134 205 291 342 428 492 606 717 762 862 966 1018 1089 1206 1235 1297 1393 1533 15891700

61 135 206 292 343 429 493 607 718 763 863 967 1019 1090 1207 1236 1298 1394 1534 15901621

62 136 207 293 344 430 494 608 719 764 864 968 1020 1091 1208 1237 1299 1395 1535 15911622

63 137 208 294 345 431 495 609 720 765 865 969 1021 1092 1209 1238 1300 1396 1536 15921623

64 138 209 295 346 432 496 610 721 766 866 970 1022 1093 1210 1239 1301 1397 1537 15931624

65 139 210 296 347 433 497 611 722 767 867 971 1023 1094 1211 1240 1302 1398 1538 15941625

66 140 211 297 348 434 498 612 723 768 868 892 1024 1095 1212 1241 1303 1399 1459 15951626

67 141 212 298 349 435 499 613 724 769 869 893 1025 1096 1213 1242 1304 1400 1460 15961627

68 142 213 299 350 436 500 614 725 770 870 894 1026 1097 1214 1243 1305 1401 1461 15971628

69 143 214 300 351 437 501 615 726 771 871 895 1027 1098 1135 1244 1306 1402 1462 15981629

70 144 215 301 352 438 502 616 727 772 872 896 1028 1099 1136 1245 1307 1403 1463 15991630

71 145 216 302 353 439 503 617 728 773 873 897 1029 1100 1137 1246 1308 1404 1464 16001631

72 146 217 303 354 440 504 618 649 774 874 898 1030 1101 1138 1247 1309 1405 1465 16011632

73 147 218 304 355 441 505 619 650 775 875 899 1031 1102 1139 1248 1310 1406 1466 16021633

74 148 219 305 356 442 506 620 651 776 876 900 1032 1103 1140 1249 1311 1407 1467 16031634

75 149 220 306 357 443 507 621 652 777 877 901 1033 1104 1141 1250 1312 1408 1468 16041635

76 150 221 307 358 444 508 622 653 778 878 902 1034 1105 1142 1251 1313 1409 1469 16051636

77 151 222 308 359 445 509 623 654 779 879 903 1035 1106 1143 1252 1314 1410 1470 16061637

78 152 223 309 360 446 510 624 655 780 880 904 1036 1107 1144 1253 1315 1411 1471 16071638

79 153 224 310 361 447 511 625 656 781 881 905 1037 1108 1145 1254 1316 1412 1472 16081639

80 154 225 311 362 448 512 626 657 782 882 906 1038 1109 1146 1255 1317 1413 1473 16091640

1 155 226 312 363 449 513 627 658 783 883 907 1039 1110 1147 1256 1318 1414 1474 16101641

2 156 227 313 364 450 514 628 659 784 884 908 1040 1111 1148 1257 1319 1415 1475 16111642

3 157 228 314 365 451 515 629 660 785 885 909 1041 1112 1149 1258 1320 1416 1476 16121643

4 158 229 315 366 452 516 630 661 786 886 910 1042 1113 1150 1259 1321 1417 1477 16131644

5 159 230 316 367 453 517 631 662 787 887 911 1043 1114 1151 1260 1322 1418 1478 16141645

6 160 231 317 368 454 518 632 663 788 888 912 1044 1115 1152 1261 1323 1419 1479 16151646

7 161 232 318 369 455 519 633 664 789 889 913 1045 1116 1153 1262 1324 1420 1480 16161647

8 82 233 319 370 456 520 634 665 790 890 914 1046 1117 1154 1263 1325 1421 1481 16171648

9 83 234 320 371 457 521 635 666 791 811 915 1047 1118 1155 1264 1326 1422 1482 16181649

10 84 23 532 137 245 852 263 666 779 281 291 610 481 119 1156 1265 1327 1423 1483 16191650

11 85 236 322 373 459 523 637 668 793 813 917 1049 1120 1157 1266 1328 1424 1484 15401651

12 86 237 323 374 460 524 638 669 794 814 918 1050 1121 1158 1267 1329 1425 1485 15411652

13 87 238 244 375 461 525 639 670 795 815 919 1051 1122 1159 1268 1330 1426 1486 15421653

21 154 226 245 400 430 507 622 712 791 843 949 982 1069 1209 1258 1362 1424 1533 15971637 1704

39 93 187 285 393 476 555 598 675 785 887 955 1042 1077 1196 1290 1318 1423 1498 15511639 1772

7 136 165 298 384 446 537 590 648 783 851 952 1050 1067 1175 1217 1302 1437 1514 15881693 1755

10 132 208 269 353 475 502 572 716 736 841 913 1033 1062 1145 1219 1368 1418 1490 15481685 1722

30 156 204 244 374 411 562 617 689 770 836 957 1000 1133 1174 1277 1365 1416 1502 15521667 1770

17 130 228 258 330 467 514 597 724 788 827 947 1048 1102 1190 1240 1326 1449 1468 15671632 1773

43 122 202 319 349 479 524 629 713 781 876 942 1051 1123 1166 1236 1370 1403 1496 15841692 1713

73 104 194 317 358 445 530 639 658 743 817 933 991 1079 1178 1260 1360 1421 1458 15631644 1780

61 149 176 270 385 473 541 645 723 750 853 902 978 1098 1144 1234 1355 1414 1537 16131654 1726

74 129 221 304 362 405 498 576 695 752 872 923 1004 1107 1172 1226 1346 1456 1510 15621660 1756

45 161 201 322 367 434 538 613 726 730 881 959 1034 1054 1134 1288 1315 1383 1465 15781671 1744

20 91 233 315 325 407 542 573 668 773 828 898 1022 1111 1213 1253 1336 1385 1534 15541628 1757

34 97 163 277 348 442 557 577 652 769 885 907 1035 1116 1186 1233 1372 1443 1479 15661668 1728

15 108 169 284 347 431 494 592 718 793 890 934 1006 1074 1141 1265 1311 1406 1464 16121672 1703

13 145 180 286 341 456 553 609 715 767 848 911 981 1097 1210 1275 1320 1402 1516 15601687 1717

46 105 217 264 380 441 523 588 676 759 871 916 995 1096 1155 1281 1347 1426 1467 15391624 1778

80 109 177 307 388 413 552 638 720 741 870 954 976 1090 1140 1292 1324 1400 1489 16011683 1776

18 124 181 303 371 444 488 587 710 786 864 897 974 1129 1192 1249 1329 1392 1473 15741653 1729

11 141 196 247 338 466 544 603 705 766 823 896 1007 1057 1143 1289 1367 1454 1459 16091682 1763

53 120 213 301 386 450 556 579 696 798 831 970 1041 1120 1165 1293 1310 1419 1536 15981698 1781

60 90 192 293 389 436 522 591 665 808 814 929 979 1087 1149 1228 1309 1399 1497 15431674 1774

62 119 242 275 329 433 550 637 686 734 861 937 1052 1055 1135 1245 1303 1431 1461 16081686 1736

40 135 191 320 396 474 486 585 722 745 829 920 1014 1058 1212 1224 1342 1441 1531 15801652 1743

3 111 207 300 342 438 511 567 661 782 832 967 1021 1078 1173 1274 1350 1447 1526 16111680 1745

79 123 183 252 372 428 564 626 670 742 852 935 1023 1065 1137 1223 1333 1378 1517 15531620 1723

23 89 195 262 360 423 519 599 697 746 839 938 1001 1091 1207 1239 1300 1415 1486 16171641 1766

77 117 241 268 373 414 508 634 674 761 865 958 1044 1121 1202 1295 1348 1455 1507 16031694 1762

69 81 189 279 344 463 533 623 679 778 815 945 1040 1109 1193 1227 1351 1379 1463 16001649 1706

51 158 162 316 399 484 518 568 717 757 883 971 984 1122 1162 1273 1371 1394 1482 15611638 1760

16 131 230 276 333 440 490 633 660 807 816 921 1038 1093 1183 1221 1358 1411 1491 16051663 1752

76 86 203 280 394 459 521 605 659 756 867 909 1030 1068 1139 1215 1304 1390 1518 15951648 1734

28 155 238 295 392 468 543 636 653 772 842 922 1012 1082 1158 1262 1334 1440 1495 15901700 1779

38 100 227 312 345 415 527 578 692 748 856 893 977 1063 1167 1235 1322 1389 1500 15811651 1759

44 85 172 291 379 472 513 642 700 760 837 948 1037 1061 1194 1270 1335 1405 1538 15501673 1711

55 137 237 261 397 477 510 628 683 806 835 962 989 1094 1171 1259 1306 1381 1481 15711657 1721

12 88 209 290 390 435 551 625 650 754 868 943 999 1128 1176 1284 1361 1393 1480 16071643 1727

52 110 240 306 352 458 515 586 698 729 822 941 1005 1066 1214 1269 1375 1439 1474 15461640 1738

56 94 182 282 359 457 505 630 701 795 840 894 1016 1059 1157 1241 1356 1387 1513 15551681 1775

71 160 166 294 361 451 500 620 721 768 833 928 973 1101 1156 1272 1354 1377 1521 15821645 1735

8 157 232 260 339 410 491 615 708 803 875 946 1013 1108 1150 1294 1307 1428 1504 15591635 1739

67 118 229 288 382 418 540 606 654 792 882 939 1017 1110 1189 1278 1341 1401 1471 15641630 1754

37 82 190 243 378 481 561 575 684 737 884 901 1032 1088 1197 1264 1359 1436 1519 16021621 1771

66 152 234 249 402 448 517 596 672 802 862 908 1049 1131 1180 1261 1352 1425 1522 15451670 1750

2 147 224 302 376 416 536 632 685 774 825 910 1028 1127 1147 1222 1314 1450 1462 15441691 1720

58 138 219 257 368 419 545 571 656 805 821 968 998 1071 1195 1266 1321 1435 1529 16181647 1749

70 107 210 246 350 439 492 580 711 747 845 931 1027 1125 1198 1256 1323 1407 1475 15771666 1765

36 128 179 271 395 426 549 607 725 731 819 927 1043 1117 1138 1251 1301 1438 1505 15851675 1741

64 84 200 256 375 452 554 584 706 797 811 951 1019 1099 1205 1242 1344 1380 1493 15681622 1753

0 103 236 308 327 482 512 589 704 794 873 925 1031 1064 1151 1291 1340 1444 1506 16151679 1719

25 112 175 259 337 470 535 627 657 755 838 917 997 1124 1181 1232 1364 1430 1477 15831684 1747

78 139 184 281 343 483 534 570 691 799 818 899 1025 1076 1169 1268 1338 1427 1532 15861642 1701

33 116 211 265 354 454 528 569 709 789 850 944 972 1086 1182 1287 1330 1388 1466 16061665 1708

22 121 188 251 391 429 487 643 702 784 860 924 986 1092 1153 1216 1312 1432 1527 15931664 1761

47 159 193 248 351 443 495 602 664 775 866 956 1039 1103 1208 1243 1357 1422 1525 16191658 1716

32 102 231 289 355 424 558 610 671 744 877 966 994 1060 1142 1220 1337 1417 1478 15691697 1705

4 101 174 253 370 422 525 593 673 765 834 892 983 1100 1203 1225 1369 1408 1512 15571625 1730

35 95 173 323 387 455 493 640 651 801 874 903 1008 1104 1201 1263 1299 1457 1530 15701688 1715

57 134 167 318 366 409 496 608 694 740 878 940 993 1119 1154 1286 1305 1398 1523 15411655 1767

41 142 206 309 336 427 516 611 690 749 813 900 1045 1056 1188 1285 1316 1434 1485 15961623 1718

27 125 214 278 365 420 503 631 714 776 830 904 996 1115 1206 1279 1353 1453 1492 16101626 1740

24 92 197 299 381 462 529 618 688 753 889 919 1018 1085 1199 1238 1313 1382 1494 15911646 1724

65 140 164 255 357 469 559 644 680 758 859 936 1002 1114 1161 1246 1317 1409 1472 15891633 1710

29 143 212 274 369 471 547 594 662 796 888 915 988 1130 1168 1229 1332 1386 1515 15421659 1707

19 83 215 283 335 449 560 582 707 739 824 965 985 1106 1170 1276 1349 1391 1511 15761689 1748

14 150 235 310 363 412 531 595 687 738 880 914 1026 1118 1148 1244 1328 1429 1535 15941677 1712

5 96 222 287 324 408 506 646 719 732 812 930 990 1084 1191 1247 1298 1452 1509 15871690 1702

54 126 168 292 404 432 520 621 649 771 858 906 980 1112 1187 1267 1327 1451 1501 15491661 1777

75 114 198 250 377 406 501 635 655 779 886 918 975 1053 1211 1254 1343 1445 1483 15561636 1768

31 127 186 273 332 478 499 616 666 762 810 964 1046 1073 1185 1280 1319 1404 1528 15581650 1737

50 98 199 272 401 460 532 614 703 809 847 912 1015 1126 1177 1230 1331 1412 1508 16161631 1758

59 153 170 266 346 425 566 647 663 777 820 891 1036 1081 1159 1218 1297 1395 1460 15791629 1714

6 87 225 305 331 485 504 601 667 780 855 953 992 1070 1204 1231 1325 1442 1470 15751662 1733

63 148 239 313 383 437 497 619 682 800 844 926 1011 1095 1184 1282 1296 1410 1476 15991696 1742

68 146 220 296 334 447 539 612 699 787 869 961 1020 1080 1136 1257 1366 1413 1487 15731634 1769

26 99 218 263 356 453 546 574 678 733 854 950 1047 1132 1146 1271 1339 1433 1524 15651627 1746

49 133 171 311 340 464 548 581 728 763 826 895 1024 1083 1152 1252 1374 1420 1484 15471669 1751

48 151 205 314 326 421 526 583 677 751 857 960 1029 1105 1163 1250 1363 1446 1488 15921676 1709

42 144 223 254 403 461 489 641 693 764 879 932 987 1089 1200 1283 1308 1396 1503 15721678 1732

1 106 216 321 364 465 565 604 669 735 863 963 1010 1075 1160 1237 1373 1384 1520 16041656 1731

9 113 178 267 328 480 509 600 681 790 849 905 1009 1072 1164 1255 1345 1397 1499 16141699 1725

72 115 185 297 398 417 563 624 727 804 846 969 1003 1113 1179 1248 1376 1448 1469 15401695 1764

23 156 228 247 402 432 509 624 714 793 845 951 984 1071 1211 1260 1364 1426 1535 15991706 1792

11 115 180 269 330 482 511 602 683 792 851 907 1011 1074 1166 1257 1347 1399 1501 16161727 1823

74 117 187 299 400 419 565 626 649 806 848 971 1005 1115 1181 1250 1298 1450 1471 15421766 1827

41 95 189 287 395 478 557 600 677 787 889 957 1044 1079 1198 1292 1320 1425 1500 15531774 1842

9 138 167 300 386 448 539 592 648 785 853 954 1052 1069 1177 1219 1304 1439 1516 15901757 1859

12 134 210 271 355 477 504 574 718 738 843 915 1035 1064 1147 1221 1370 1420 1492 15501724 1838

32 158 206 246 376 413 564 619 691 772 838 959 1002 1055 1176 1279 1367 1418 1504 15541772 1808

19 132 230 260 332 469 516 599 726 790 829 949 1050 1104 1192 1242 1328 1451 1470 15691775 1837

45 124 204 321 351 481 526 631 715 783 878 944 973 1125 1168 1238 1372 1405 1498 15861715 1853

75 106 196 319 360 447 532 641 660 745 819 935 993 1081 1180 1262 1362 1423 1458 15651702 1829

63 151 178 272 387 475 543 647 725 752 855 904 980 1100 1146 1236 1357 1416 1459 16151728 1841

76 131 223 306 364 405 500 578 697 754 874 925 1006 1109 1174 1228 1348 1378 1512 15641758 1807

47 83 203 244 369 436 540 615 728 732 883 961 1036 1056 1134 1290 1317 1385 1467 15801746 1835

22 93 235 317 327 409 544 575 670 775 830 900 1024 1113 1135 1255 1338 1387 1536 15561759 1782

36 99 165 279 350 444 559 579 654 771 887 909 1037 1118 1188 1235 1374 1445 1481 15681730 1796

17 110 171 286 349 433 496 594 720 795 812 936 1008 1076 1143 1267 1313 1408 1466 16141705 1849

15 147 182 288 343 458 555 611 717 769 850 913 983 1099 1212 1277 1322 1404 1518 15621719 1804

48 107 219 266 382 443 525 590 678 761 873 918 997 1098 1157 1283 1349 1428 1469 15391780 1793

2 111 179 309 390 415 554 640 722 743 872 956 978 1092 1142 1294 1326 1402 1491 16031778 1818

20 126 183 305 373 446 490 589 712 788 866 899 976 1131 1194 1251 1331 1394 1475 15761731 1803

13 143 198 249 340 468 546 605 707 768 825 898 1009 1059 1145 1291 1369 1456 1461 16111765 1855

55 122 215 303 388 452 558 581 698 800 833 892 1043 1122 1167 1295 1312 1421 1538 16001703 1806

62 92 194 295 391 438 524 593 667 730 816 931 981 1089 1151 1230 1311 1401 1499 15451776 1828

64 121 164 277 331 435 552 639 688 736 863 939 974 1057 1137 1247 1305 1433 1463 16101738 1812

42 137 193 322 398 476 486 587 724 747 831 922 1016 1060 1214 1226 1344 1443 1533 15821745 1798

5 113 209 302 344 440 513 567 663 784 834 969 1023 1080 1175 1276 1352 1449 1528 16131747 1795

1 125 185 254 374 430 566 628 672 744 854 937 1025 1067 1139 1225 1335 1380 1519 15551725 1836

25 91 197 264 362 425 521 601 699 748 841 940 1003 1093 1209 1241 1302 1417 1488 16191768 1800

79 119 163 270 375 416 510 636 676 763 867 960 1046 1123 1204 1217 1350 1457 1509 16051764 1790

71 81 191 281 346 465 535 625 681 780 817 947 1042 1111 1195 1229 1353 1381 1465 16021708 1785

53 160 162 318 401 406 520 570 719 759 885 893 986 1124 1164 1275 1373 1396 1484 15631762 1856

18 133 232 278 335 442 492 635 662 809 818 923 1040 1095 1185 1223 1360 1413 1493 16071754 1825

78 88 205 282 396 461 523 607 661 758 869 911 1032 1070 1141 1215 1306 1392 1520 15971736 1846

30 157 240 297 394 470 545 638 655 774 844 924 1014 1084 1160 1264 1336 1442 1497 15921781 1802

40 102 229 314 347 417 529 580 694 750 858 895 979 1065 1169 1237 1324 1391 1502 15831761 1821

46 87 174 293 381 474 515 644 702 762 839 950 1039 1063 1196 1272 1337 1407 1460 15521713 1830

57 139 239 263 399 479 512 630 685 808 837 964 991 1096 1173 1261 1308 1383 1483 15731723 1857

14 90 211 292 392 437 553 627 652 756 870 945 1001 1130 1178 1286 1363 1395 1482 16091729 1834

54 112 242 308 354 460 517 588 700 729 824 943 1007 1068 1136 1271 1297 1441 1476 15481740 1839

58 96 184 284 361 459 507 632 703 797 842 896 1018 1061 1159 1243 1358 1389 1515 15571777 1797

73 82 168 296 363 453 502 622 723 770 835 930 975 1103 1158 1274 1356 1377 1523 15841737 1820

10 159 234 262 341 412 493 617 710 805 877 948 1015 1110 1152 1216 1309 1430 1506 15611741 1819

69 120 231 290 384 420 542 608 656 794 884 941 1019 1112 1191 1280 1343 1403 1473 15661756 1813

39 84 192 243 380 483 563 577 686 739 886 903 1034 1090 1199 1266 1361 1438 1521 16041773 1852

68 154 236 251 404 450 519 598 674 804 864 910 1051 1133 1182 1263 1354 1427 1524 15471752 1860

4 149 226 304 378 418 538 634 687 776 827 912 1030 1129 1149 1224 1316 1452 1464 15461722 1843

60 140 221 259 370 421 547 573 658 807 823 970 1000 1073 1197 1268 1323 1437 1531 15401751 1810

72 109 212 248 352 441 494 582 713 749 847 933 1029 1127 1200 1258 1325 1409 1477 15791767 1858

38 130 181 273 397 428 551 609 727 733 821 929 1045 1119 1140 1253 1303 1440 1507 15871743 1861

66 86 202 258 377 454 556 586 708 799 813 953 1021 1101 1207 1244 1346 1382 1495 15701755 1801

0 105 238 310 329 484 514 591 706 796 875 927 1033 1066 1153 1293 1342 1446 1508 16171721 1788

27 114 177 261 339 472 537 629 659 757 840 919 999 1126 1183 1234 1366 1432 1479 15851749 1814

80 141 186 283 345 485 536 572 693 801 820 901 1027 1078 1171 1270 1340 1429 1534 15881701 1844

35 118 213 267 356 456 530 571 711 791 852 946 972 1088 1184 1289 1332 1390 1468 16081710 1832

24 123 190 253 393 431 489 645 704 786 862 926 988 1094 1155 1218 1314 1434 1529 15951763 1845

49 161 195 250 353 445 497 604 666 777 868 958 1041 1105 1210 1245 1359 1424 1527 15411718 1816

34 104 233 291 357 426 560 612 673 746 879 968 996 1062 1144 1222 1339 1419 1480 15711707 1791

6 103 176 255 372 424 527 595 675 767 836 894 985 1102 1205 1227 1371 1410 1514 15591732 1805

37 97 175 245 389 457 495 642 653 803 876 905 1010 1106 1203 1265 1301 1379 1532 15721717 1786

59 136 169 320 368 411 498 610 696 742 880 942 995 1121 1156 1288 1307 1400 1525 15431769 1784

43 144 208 311 338 429 518 613 692 751 815 902 1047 1058 1190 1287 1318 1436 1487 15981720 1817

29 127 216 280 367 422 505 633 716 778 832 906 998 1117 1208 1281 1355 1455 1494 16121742 1851

26 94 199 301 383 464 531 620 690 755 811 921 1020 1087 1201 1240 1315 1384 1496 15931726 1789

67 142 166 257 359 471 561 646 682 760 861 938 1004 1116 1163 1248 1319 1411 1474 15911712 1862

31 145 214 276 371 473 549 596 664 798 890 917 990 1132 1170 1231 1334 1388 1517 15441709 1824

21 85 217 285 337 451 562 584 709 741 826 967 987 1108 1172 1278 1351 1393 1513 15781750 1831

16 152 237 312 365 414 533 597 689 740 882 916 1028 1120 1150 1246 1330 1431 1537 15961714 1833

7 98 224 289 324 410 508 568 721 734 814 932 992 1086 1193 1249 1300 1454 1511 15891704 1811

56 128 170 294 326 434 522 623 651 773 860 908 982 1114 1189 1269 1329 1453 1503 15511779 1854

77 116 200 252 379 408 503 637 657 781 888 920 977 1053 1213 1256 1345 1447 1485 15581770 1850

33 129 188 275 334 480 501 618 668 764 810 966 1048 1075 1187 1282 1321 1406 1530 15601739 1794

52 100 201 274 403 462 534 616 705 731 849 914 1017 1128 1179 1232 1333 1414 1510 16181760 1848

61 155 172 268 348 427 488 569 665 779 822 891 1038 1083 1161 1220 1299 1397 1462 15811716 1840

8 89 227 307 333 407 506 603 669 782 857 955 994 1072 1206 1233 1327 1444 1472 15771735 1822

65 150 241 315 385 439 499 621 684 802 846 928 1013 1097 1186 1284 1296 1412 1478 16011744 1787

70 148 222 298 336 449 541 614 701 789 871 963 1022 1082 1138 1259 1368 1415 1489 15751771 1847

28 101 220 265 358 455 548 576 680 735 856 952 1049 1054 1148 1273 1341 1435 1526 15671748 1799

51 135 173 313 342 466 550 583 650 765 828 897 1026 1085 1154 1254 1376 1422 1486 15491753 1809

50 153 207 316 328 423 528 585 679 753 859 962 1031 1107 1165 1252 1365 1448 1490 15941711 1815

44 146 225 256 325 463 491 643 695 766 881 934 989 1091 1202 1285 1310 1398 1505 15741734 1826

3 108 218 323 366 467 487 606 671 737 865 965 1012 1077 1162 1239 1375 1386 1522 16061733 1783

25 158 230 249 404 434 511 626 716 795 847 953 986 1073 1213 1262 1366 1428 1537 16011794 1863

46 148 227 258 327 465 493 645 697 768 883 936 991 1093 1204 1287 1312 1400 1507 15761828 1905

5 110 220 245 368 469 489 608 673 739 867 967 1014 1079 1164 1241 1297 1388 1524 16081785 1906

13 117 182 271 332 484 513 604 685 794 853 909 1013 1076 1168 1259 1349 1401 1503 16181825 1907

76 119 189 301 402 421 487 628 651 808 850 893 1007 1117 1183 1252 1300 1452 1473 15441829 1908

43 97 191 289 397 480 559 602 679 789 811 959 1046 1081 1200 1294 1322 1427 1502 15551844 1909

11 140 169 302 388 450 541 594 648 787 855 956 974 1071 1179 1221 1306 1441 1518 15921861 1910

14 136 212 273 357 479 506 576 720 740 845 917 1037 1066 1149 1223 1372 1422 1494 15521840 1911

34 160 208 248 378 415 566 621 693 774 840 961 1004 1057 1178 1281 1369 1420 1506 15561810 1912

21 134 232 262 334 471 518 601 728 792 831 951 1052 1106 1194 1244 1330 1453 1472 15711839 1913

47 126 206 323 353 483 528 633 717 785 880 946 975 1127 1170 1240 1374 1407 1500 15881855 1914

77 108 198 321 362 449 534 643 662 747 821 937 995 1083 1182 1264 1364 1425 1458 15671831 1915

65 153 180 274 389 477 545 569 727 754 857 906 982 1102 1148 1238 1359 1418 1461 16171843 1916

78 133 225 308 366 405 502 580 699 756 876 927 1008 1111 1176 1230 1350 1380 1514 15661809 1917

49 85 205 246 371 438 542 617 650 734 885 963 1038 1058 1134 1292 1319 1387 1469 15821837 1918

24 95 237 319 329 411 546 577 672 777 832 902 1026 1115 1137 1257 1340 1389 1538 15581782 1919

38 101 167 281 352 446 561 581 656 773 889 911 1039 1120 1190 1237 1376 1447 1483 15701798 1920

19 112 173 288 351 435 498 596 722 797 814 938 1010 1078 1145 1269 1315 1410 1468 16161851 1921

17 149 184 290 345 460 557 613 719 771 852 915 985 1101 1214 1279 1324 1406 1520 15641806 1922

50 109 221 268 384 445 527 592 680 763 875 920 999 1100 1159 1285 1351 1430 1471 15391795 1923

4 113 181 311 392 417 556 642 724 745 874 958 980 1094 1144 1216 1328 1404 1493 16051820 1924

22 128 185 307 375 448 492 591 714 790 868 901 978 1133 1196 1253 1333 1396 1477 15781805 1925

15 145 200 251 342 470 548 607 709 770 827 900 1011 1061 1147 1293 1371 1378 1463 16131857 1926

57 124 217 305 390 454 560 583 700 802 835 894 1045 1124 1169 1217 1314 1423 1460 16021808 1927

64 94 196 297 393 440 526 595 669 732 818 933 983 1091 1153 1232 1313 1403 1501 15471830 1928

66 123 166 279 333 437 554 641 690 738 865 941 976 1059 1139 1249 1307 1435 1465 16121814 1929

44 139 195 244 400 478 486 589 726 749 833 924 1018 1062 1136 1228 1346 1445 1535 15841800 1930

7 115 211 304 346 442 515 567 665 786 836 971 1025 1082 1177 1278 1354 1451 1530 16151797 1931

3 127 187 256 376 432 488 630 674 746 856 939 1027 1069 1141 1227 1337 1382 1521 15571838 1932

27 93 199 266 364 427 523 603 701 750 843 942 1005 1095 1211 1243 1304 1419 1490 15411802 1933

1 121 165 272 377 418 512 638 678 765 869 962 1048 1125 1206 1219 1352 1379 1511 16071792 1934

73 81 193 283 348 467 537 627 683 782 819 949 1044 1113 1197 1231 1355 1383 1467 16041787 1935

55 82 162 320 403 408 522 572 721 761 887 895 988 1126 1166 1277 1375 1398 1486 15651858 1936

20 135 234 280 337 444 494 637 664 731 820 925 1042 1097 1187 1225 1362 1415 1495 16091827 1937

80 90 207 284 398 463 525 609 663 760 871 913 1034 1072 1143 1215 1308 1394 1522 15991848 1938

32 159 242 299 396 472 547 640 657 776 846 926 1016 1086 1162 1266 1338 1444 1499 15941804 1939

42 104 231 316 349 419 531 582 696 752 860 897 981 1067 1171 1239 1326 1393 1504 15851823 1940

48 89 176 295 383 476 517 646 704 764 841 952 1041 1065 1198 1274 1339 1409 1462 15541832 1941

59 141 241 265 401 481 514 632 687 730 839 966 993 1098 1175 1263 1310 1385 1485 15751859 1942

16 92 213 294 394 439 555 629 654 758 872 947 1003 1132 1180 1288 1365 1397 1484 16111836 1943

56 114 164 310 356 462 519 590 702 729 826 945 1009 1070 1138 1273 1299 1443 1478 15501841 1864

60 98 186 286 363 461 509 634 705 799 844 898 1020 1063 1161 1245 1360 1391 1517 15591799 1865

75 84 170 298 365 455 504 624 725 772 837 932 977 1105 1160 1276 1358 1377 1525 15861822 1866

12 161 236 264 343 414 495 619 712 807 879 950 1017 1112 1154 1218 1311 1432 1508 15631821 1867

71 122 233 292 386 422 544 610 658 796 886 943 1021 1114 1193 1282 1345 1405 1475 15681815 1868

41 86 194 243 382 485 565 579 688 741 888 905 1036 1092 1201 1268 1363 1440 1523 16061854 1869

70 156 238 253 326 452 521 600 676 806 866 912 973 1055 1184 1265 1356 1429 1526 15491862 1870

6 151 228 306 380 420 540 636 689 778 829 914 1032 1131 1151 1226 1318 1454 1466 15481845 1871

62 142 223 261 372 423 549 575 660 809 825 892 1002 1075 1199 1270 1325 1439 1533 15421812 1872

74 111 214 250 354 443 496 584 715 751 849 935 1031 1129 1202 1260 1327 1411 1479 15811860 1873

40 132 183 275 399 430 553 611 649 735 823 931 1047 1121 1142 1255 1305 1442 1509 15891783 1874

68 88 204 260 379 456 558 588 710 801 815 955 1023 1103 1209 1246 1348 1384 1497 15721803 1875

0 107 240 312 331 406 516 593 708 798 877 929 1035 1068 1155 1295 1344 1448 1510 16191790 1876

29 116 179 263 341 474 539 631 661 759 842 921 1001 1128 1185 1236 1368 1434 1481 15871816 1877

2 143 188 285 347 407 538 574 695 803 822 903 1029 1080 1173 1272 1342 1431 1536 15901846 1878

37 120 215 269 358 458 532 573 713 793 854 948 972 1090 1186 1291 1334 1392 1470 16101834 1879

26 125 192 255 395 433 491 647 706 788 864 928 990 1096 1157 1220 1316 1436 1531 15971847 1880

51 83 197 252 355 447 499 606 668 779 870 960 1043 1107 1212 1247 1361 1426 1529 15431818 1881

36 106 235 293 359 428 562 614 675 748 881 970 998 1064 1146 1224 1341 1421 1482 15731793 1882

8 105 178 257 374 426 529 597 677 769 838 896 987 1104 1207 1229 1373 1412 1516 15611807 1883

39 99 177 247 391 459 497 644 655 805 878 907 1012 1108 1205 1267 1303 1381 153415741788 1884

61 138 171 322 370 413 500 612 698 744 882 944 997 1123 1158 1290 1309 1402 1527 15451786 1885

45 146 210 313 340 431 520 615 694 753 817 904 1049 1060 1192 1289 1320 1438 1489 16001819 1886

31 129 218 282 369 424 507 635 718 780 834 908 1000 1119 1210 1283 1357 1457 1496 16141853 1887

28 96 201 303 385 466 533 622 692 757 813 923 1022 1089 1203 1242 1317 1386 1498 15951791 1888

69 144 168 259 361 473 563 568 684 762 863 940 1006 1118 1165 1250 1321 1413 1476 15931784 1889

33 147 216 278 373 475 551 598 666 800 812 919 992 1054 1172 1233 1336 1390 1519 15461826 1890

23 87 219 287 339 453 564 586 711 743 828 969 989 1110 1174 1280 1353 1395 1515 15801833 1891

18 154 239 314 367 416 535 599 691 742 884 918 1030 1122 1152 1248 1332 1433 1459 15981835 1892

9 100 226 291 324 412 510 570 723 736 816 934 994 1088 1195 1251 1302 1456 1513 15911813 1893

58 130 172 296 328 436 524 625 653 775 862 910 984 1116 1191 1271 1331 1455 1505 15531856 1894

79 118 202 254 381 410 505 639 659 783 890 922 979 1053 1135 1258 1347 1449 1487 15601852 1895

35 131 190 277 336 482 503 620 670 766 810 968 1050 1077 1189 1284 1323 1408 1532 15621796 1896

54 102 203 276 325 464 536 618 707 733 851 916 1019 1130 1181 1234 1335 1416 1512 15401850 1897

63 157 174 270 350 429 490 571 667 781 824 891 1040 1085 1163 1222 1301 1399 1464 15831842 1898

10 91 229 309 335 409 508 605 671 784 859 957 996 1074 1208 1235 1329 1446 1474 15791824 1899

67 152 163 317 387 441 501 623 686 804 848 930 1015 1099 1188 1286 1296 1414 1480 16031789 1900

72 150 224 300 338 451 543 616 703 791 873 965 1024 1084 1140 1261 1370 1417 1491 15771849 1901

30 103 222 267 360 457 550 578 682 737 858 954 1051 1056 1150 1275 1343 1437 1528 15691801 1902

53 137 175 315 344 468 552 585 652 767 830 899 1028 1087 1156 1256 1298 1424 1488 15511811 1903

52 155 209 318 330 425 530 587 681 755 861 964 1033 1109 1167 1254 1367 1450 1492 15961817 1904

Claims (8)

1, a kind of method that makes up based on the GRS abnormal LDPC code is characterized in that described method comprises:

For the LDPC code block is selected a plurality of possible bit degree distribution;

Select to obtain a bit degree distribution of the top performance limit in described a plurality of possible bit degree distribution based on the performance of determining by the density evolution method;

Based on selected bit degree distribution, will be decomposed into a plurality of submatrixs corresponding to parity matrix based on GRS rule LDPC sign indicating number, each submatrix in wherein said a plurality of submatrixs has corresponding bit degree, and has corresponding a plurality of permutation matrixes; And

Replace at least one permutation matrix at least one submatrix in described a plurality of submatrix so that at least one submatrix includes null matrix and permutation matrix with null matrix, thereby generate corresponding based on the GRS abnormal LDPC code.

2, method according to claim 1 is characterized in that, described method further comprises:

Each bit degree distribution in described a plurality of possible bit degree distribution is carried out the analysis of density evolution method, thereby generate the density evolution method analysis result of a plurality of correspondences; And

By more described density evolution method analysis result, from described a plurality of possible bit degree distribution, select a bit degree distribution.

3, method according to claim 1 is characterized in that, described a plurality of submatrixs are three submatrixs.

4, method according to claim 3 is characterized in that, described method further comprises:

Replace at least one permutation matrix in first submatrix in described three submatrixs with null matrix;

Replace in described three submatrixs at least one permutation matrix in second submatrix with null matrix.

5, method according to claim 1 is characterized in that, described method further comprises:

Generation and at least the second is based on corresponding at least the second parity matrix of GRS abnormal LDPC code;

To being applied to described in the communication channel, determine in first performance aspect the Block Error Rate of the function of signal to noise ratio based on the GRS abnormal LDPC code;

To being applied to described at least the second in the described communication channel, determine in second performance aspect the Block Error Rate of the function of signal to noise ratio based on the GRS abnormal LDPC code;

When described first performance has than the lower wrong rank of described second performance, select described based on the GRS abnormal LDPC code;

When described second performance has than the lower wrong rank of described first performance, select described at least the second based on the GRS abnormal LDPC code.

6, a kind of method of selecting based on the GRS abnormal LDPC code is characterized in that described method comprises:

Make up with a plurality of based on the corresponding a plurality of parity matrixs of GRS abnormal LDPC code;

Determine that in a plurality of performances aspect the Block Error Rate of the function of signal to noise ratio described a plurality of performances are corresponding with described a plurality of each based on the GRS abnormal LDPC code that are used for communication channel;

From described a plurality of based on selecting to have of the lowest error rank determined according to described a plurality of performances the GRS abnormal LDPC code based on the GRS abnormal LDPC code, described selection have corresponding parity matrix based on the GRS abnormal LDPC code.

7, method according to claim 6 is characterized in that, described method further comprises:

Consideration will be used for the complexity according to described a plurality of first first decoders of decoding based on GRS abnormal LDPC code encoded signals based on the GRS abnormal LDPC code;

Consideration will be used for the complexity according to described a plurality of second second decoders of decoding based on GRS abnormal LDPC code encoded signals based on the GRS abnormal LDPC code;

When the complexity of described first decoder is lower than described second decoder, select described first based on the GRS abnormal LDPC code;

When the complexity of described second decoder is lower than described first decoder, select described second based on the GRS abnormal LDPC code.

8, method according to claim 6 is characterized in that, the parity matrix in described a plurality of parity matrixs comprises first submatrix and second submatrix; At least one permutation matrix in described second submatrix is replaced by null matrix.

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