CN102055485A - Quasi-cyclic low-density parity-check (QC-LDPC) code and correcting and linear coding method thereof - Google Patents

Quasi-cyclic low-density parity-check (QC-LDPC) code and correcting and linear coding method thereof Download PDF

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CN102055485A
CN102055485A CN2010106058385A CN201010605838A CN102055485A CN 102055485 A CN102055485 A CN 102055485A CN 2010106058385 A CN2010106058385 A CN 2010106058385A CN 201010605838 A CN201010605838 A CN 201010605838A CN 102055485 A CN102055485 A CN 102055485A
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吴晓富
崔龙
赵春明
尤肖虎
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PLA University of Science and Technology
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Abstract

The invention relates to a quasi-cyclic low-density parity-check (QC-LDPC) code and a correcting and linear coding method thereof. The variable nodes of the LDPC code, the dimensionality of which is greater than 2, are informational nodes; and the variable nodes the dimensionality of which is 2 form a big end-to-end ring on a bipartite graph. The correcting method comprises the following implementation steps: randomly selecting one edge on the big ring the dimensionality of which is 2, cutting off the edge, and filling 0 in the corresponding position of a low-density check matrix, thereby acquiring a corrected structure of the code. The linear coding method of the corrected code comprises the following implementation steps: firstly, multiplying an input information vector s and a part of the check matrix the load of which is greater than 2 as a vector by a matrix to acquire an intermediate vector u; directly intercepting the corresponding position of the intermediate vector u to acquire a coding vector the variable node dimensionality of which is 1; computing bit by bit from a start bit according to the characteristics of the big ring on the bipartite graph to acquire a coding vector the variable node dimensionality of which is 2; and combining the two coding vectors to finally form a coding vector.

Description

Quasi-cyclic low-density parity check code and its correcting and linear coding method
Technical Field
The invention relates to a coding method of a low-density parity check code, belonging to the field of coding structure and coding method of channel error correction coding.
Background
Among the encoding methods of Low-Density Parity-Check (LDPC) codes, Richardson proposes an encoding method based on an under-class triangular Check matrix in the literature (t.j. Richardson and r.l. Urbanke, "Efficient encoding of Low-Density Parity-Check codes," IEEE trans. info. Theory, vol.47, No.2, pp. 638-. However, if the check matrix cannot be adjusted to the ideal triangle matrix form under the class, the encoding complexity is still high. From the perspective of the hardware implementation of the LDPC codec, the structureless LDPC code is not favorable for the hardware implementation. Therefore, the LDPC code with a Quasi-Cyclic (QC) structure is widely favored by academia and industry, and the QC structure reflects the check matrix of the LDPC code and has the following characteristics: the check matrix is composed of cyclic shift sub-arrays, and the cyclic shift sub-arrays are square arrays obtained by cyclic shifting the identity matrix, so that the cyclic shift matrix is completely determined by a shift coefficient on the premise of given matrix size. Researches show that the QC structure can simplify the design of the encoder, and a plurality of encoders of QC-LDPC codes can be effectively realized by using a cyclic shift register. However, the QC-simplified encoder also has a serious constraint relationship: the check matrix must have a full rank sub-matrix of cyclic sub-matrices. In an actual configuration, this condition is not easily satisfied.
In the low-bit rate LDPC code structure, to improve performance, it is generally necessary to introduce hidden variable nodes, which is equivalent to introducing more columns (1 variable node corresponds to 1 column in the check matrix) in the check matrix of the LDPC code, that is, corresponding coded bits are not sent to a channel, and thus referred to as an LDPC code with hidden nodes, for example, a multilateral LDPC code proposed in documents (t. Richardson and r. Urbanke, "Multi-Edge type LDPC Codes," http:// lthww.epfl.ch ") or an Accumulate-Repeat-Accumulate code (referred to as" Accumulate-Accumulate Codes, "Information, intelligence, junne, June 2004) proposed in documents (a. Abbasfar, d. divsarar, and k. ao," Accumulate-Accumulate Codes, "in IEEE International syndrome on thermal).
For the convenience of implementation, the LDPC code with implicit nodes should also adopt a quasi-cyclic structure, that is, a so-called quasi-cyclic low density parity check code with implicit nodes. However, such quasi-cyclic structures often fail to find a good coding method, because the quasi-cyclic structures make it unlikely that finding a sub-matrix of information length size consisting of cyclically shifted sub-matrices in the check matrix is of full rank. Therefore, the invention adjusts the structure of the quasi-cyclic low-density parity check code with the hidden node, and provides a linear coding method based on the structure.
Disclosure of Invention
The technical problem is as follows:the invention aims to provide a correction method of a quasi-cyclic low-density parity check code and a linear coding method thereof, and solves the problem that linear complexity coding of the low-density parity check code is difficult to design.
The technical scheme is as follows:the dimensionality of the variable nodes of the quasi-cyclic low-density parity check code is divided into 3 types: dimension 1, dimension 2, and dimension greater than 2; variable nodes with dimension larger than 2 are all information nodes, correspond to information bits to be coded, and are called hidden nodes because the variable nodes are not generally sent to a channel; variable nodes with dimension 2 are just at corresponding low densityAnd a large ring connected end to end is formed on the bipartite graph corresponding to the degree check matrix.
The correction method of the quasi-cyclic low-density parity check code comprises the following steps: optionally selecting one edge on the large ring with the dimension of 2, and cutting off the edge, namely filling 0 in the corresponding position of the low-density check matrix, thereby obtaining a modified structure of the code; and filling a row in which the 0 operation is positioned in the check matrix as a position for starting coding, and calling the row as a coding starting row.
The modified structure now modifies the check matrix of the low-density parity check code, the modifying operation is that the columns in the check matrix, which form a large ring, have a column weight of 2, any "1" of one column is taken to set it to "0", the row where the replacement occurs is called a start row, and the modified structure is specifically expressed in combination with the definition of the check matrix as follows:
defining:a class of check matrices with implicit node quasi-cyclic low density parity check codes:
Figure 888274DEST_PATH_IMAGE001
wherein,is of size
Figure 550779DEST_PATH_IMAGE003
The permutation sub-matrix, which is completely dependent on the cyclic shift offset,
Figure 903263DEST_PATH_IMAGE004
for cyclically shifting the number of rows occupied by the permutated sub-matrix in the check matrix,
Figure 580232DEST_PATH_IMAGE005
permuting the number of columns occupied by the submatrix for cyclic shifts in the check matrix, the
Figure 80483DEST_PATH_IMAGE006
The size of the matrix is
Figure 766680DEST_PATH_IMAGE007
(ii) a For ease of encoding, such check matrices are divided into 3 parts:
Figure 35987DEST_PATH_IMAGE008
wherein
Figure 149436DEST_PATH_IMAGE009
information bit portion corresponding to the complete codeword of size
Figure 136984DEST_PATH_IMAGE010
(ii) a Single-dimensional check matrix
Figure 361292DEST_PATH_IMAGE011
Corresponding to a check bit portion with a single dimension column weight of 1
Figure 688368DEST_PATH_IMAGE012
(ii) a Two-dimensional check matrix
Figure 536501DEST_PATH_IMAGE013
A check bit part with a size of 2 corresponding to a two-dimensional column of a codeword
Figure 683448DEST_PATH_IMAGE014
(ii) a Total length of the code is
Figure 773764DEST_PATH_IMAGE015
(ii) a Due to the fact that
Figure 955347DEST_PATH_IMAGE016
The information bit part of the corresponding code word is not sent to the channel, and thus is a low density check code with hidden nodes; the two-dimensional check matrix
Figure 472916DEST_PATH_IMAGE017
All '1's form a large ring, and a two-dimensional check matrix is arranged
Figure 372738DEST_PATH_IMAGE017
The coordinates of the '1' in the matrix in the counterclockwise sequence of the macrocycle are sequentially
Figure 938849DEST_PATH_IMAGE018
。
The two-dimensional check matrix
Figure 37255DEST_PATH_IMAGE017
One of the '1' is arbitrarily established, the '0' is set, and the modified two-dimensional check matrix is recorded as
Figure 227190DEST_PATH_IMAGE019
And the low-density check code of the final correction structure has a check matrix:
。
the linear coding method of the quasi-cyclic low-density parity check code of the invention comprises the following steps: and calculating the coding bit by using the low-density check matrix of the modified structure and the input information bit vector: first using the input information vectorsAnd multiplying the vector with the part of the check matrix with the column number greater than 2 by the matrix to obtain an intermediate vectoru(ii) a Coding vector with variable node dimension of 1
Figure 780848DEST_PATH_IMAGE021
Direct truncation of intermediate vectorsuThe corresponding position of (2) is obtained; coding vector with variable node dimension of 2
Figure 937023DEST_PATH_IMAGE022
Then bit-by-bit calculation starting from the start bit by its macrocyclic nature on the bipartite graph will be availableThe two parts of code vectors are spliced to finally form a code output vector。
7. The linear coding method of quasi-cyclic low density parity check code according to claim 6, wherein the coding vector is divided into two parts, one part corresponding to the column with the column weight of 1 of the check matrix is obtained by directly coding the information vector; the other part corresponds to the column with the check matrix column weight of 2, the corresponding coding vector can be obtained by bit-by-bit calculation through the large ring characteristic, and the coding algorithm is specifically expressed as a plurality of steps which are executed in the following sequence:defining:let the input vector of the encoder be
Figure 467547DEST_PATH_IMAGE024
Wherein
Figure 375461DEST_PATH_IMAGE025
(ii) a The output of the encoder is a coded codeword, noted(ii) a If the information bit corresponds to the hidden node, the information bit is not transmitted and the output of the encoder is
Figure 980197DEST_PATH_IMAGE027
Whereina code vector corresponding to the one-dimensional check matrix and having a size set to；A code vector corresponding to the two-dimensional check matrix and having a size set to
Figure 745973DEST_PATH_IMAGE031
(ii) a Will matrix
Figure 63821DEST_PATH_IMAGE009
Write to a partitioned matrixWherein
Figure 626707DEST_PATH_IMAGE033
Is of a size of
Figure 936465DEST_PATH_IMAGE034
,Is of a size ofAnd is and
Figure 759431DEST_PATH_IMAGE037
；step 1:using input information bit vectors
Figure 240091DEST_PATH_IMAGE038
And a check matrix
Figure 798111DEST_PATH_IMAGE033
Multiplication direct calculation
Figure 941692DEST_PATH_IMAGE039
；Step 2Using input information bit vectorsAnd a check matrix
Figure 615436DEST_PATH_IMAGE035
Multiplication direct calculation
Figure 395173DEST_PATH_IMAGE040
；And step 3:using intermediate result vectors
Figure 790383DEST_PATH_IMAGE041
And input information bit vector in check matrix
Figure 667072DEST_PATH_IMAGE038
And a check matrixIn
Figure 818884DEST_PATH_IMAGE013
Computing a codeword vectorAs follows:
Figure 250445DEST_PATH_IMAGE044
and 4, step 4:: combining the results of the step 1 and the step 3 to finally obtain the code word。
Has the advantages that:the main innovation point of the method is that according to the characteristic that variable nodes with the dimensionality of 2 form a large ring, one variable node is selected at will on the large ring and one edge of the large ring is cut off, so that the coding can be directly calculated and completed according to the connection relation of a check matrix.
The method is mainly characterized in that:
1) compared with the original LDPC code, the LDPC code after structure modification has small change and one edge is cut off
The performance is basically unchanged, and the decoding design can still use the quasi-cyclic structure to be effective-4-
Carrying out the following steps;
2) the check matrix can be coded without Gaussian elimination change, and due to the low density characteristic of the LDPC code check matrix, the coding complexity is low.
Drawings
FIG. 1 is a large ring structure formed by submatrices with variable dimension 2 in a quasi-cyclic LDPC code check matrix.
All the symbols note:
LDPC: abbreviation of Low-sensitivity Parity-Check, Low-Density Parity-Check code;
: a check matrix of the original LDPC code;
Figure 63046DEST_PATH_IMAGE042
: a check matrix of the LDPC code after structure correction;
Figure 586431DEST_PATH_IMAGE009
: the LDPC code check matrix corresponds to a sub-matrix of information bits;
Figure 813013DEST_PATH_IMAGE011
: a check bit part corresponding to a code word in a single dimension (the column weight is 1);
Figure 54639DEST_PATH_IMAGE013
: a portion of check bits corresponding to a codeword with two dimensions (column weight 2);
Figure 657659DEST_PATH_IMAGE019
: the modified two-dimensional check matrix;
Figure 35550DEST_PATH_IMAGE046
: an input vector of an encoder;
Figure 636296DEST_PATH_IMAGE047
: the output of the encoder, encoding the codeword vector;
: a code vector corresponding to the one-dimensional check matrix;
Figure 7814DEST_PATH_IMAGE030
: a code vector corresponding to the two-dimensional check matrix;
Figure 302529DEST_PATH_IMAGE018
：
Figure 74176DEST_PATH_IMAGE017
the position of "1" in (1) is the coordinate in the matrix in the counterclockwise order of the macrocycle.
Detailed Description
The quasi-cyclic low-density parity check code of the invention selects an optional edge on the large ring with the dimension of 2, cuts off the edge, namely fills 0 in the corresponding position of the low-density check matrix, thereby obtaining a modified structure of the code. Let the check matrix of the original code be divided into 3 parts:
Figure 24814DEST_PATH_IMAGE008
and a two-dimensional check matrix
Figure 969636DEST_PATH_IMAGE017
All "1" s constitute one large ring. Is provided with
Figure 322120DEST_PATH_IMAGE017
The coordinates of the '1' in the matrix in the counterclockwise sequence of the macrocycle are sequentially
Figure 326985DEST_PATH_IMAGE018
. The structure correction method comprises the following specific steps: arbitrarily formulating two-dimensional check matrix
Figure 764920DEST_PATH_IMAGE017
One of the two-dimensional check matrixes is set to be '0', and the modified two-dimensional check matrix is recorded as
Figure 14898DEST_PATH_IMAGE019
And the final structure modified low-density check code has a check matrix:
Figure 221888DEST_PATH_IMAGE020
。
the linear coding method of quasi-cyclic low-density parity check code utilizes low-density check matrix and input information bit vector to directly calculate coding bit. First using the input information vector
Figure 335338DEST_PATH_IMAGE048
And multiplying the vector with the part of the check matrix with the column weight (namely the dimension of the variable node) larger than 2 by the matrix to obtain an intermediate vector(ii) a Coding vector with variable node dimension of 1
Figure 547194DEST_PATH_IMAGE021
Direct truncation of intermediate vectorsThe corresponding position of (2) is obtained; coding vector with variable node dimension of 2
Figure 955358DEST_PATH_IMAGE022
Then the binary code vector is calculated bit by starting from the starting bit through the large ring characteristic of the binary code vector on the bipartite graph, and the two partial code vectors are spliced to finally form the code vector
Figure 367885DEST_PATH_IMAGE023
. The linear coding method of the quasi-cyclic low density parity check code may be expressed as steps performed in the following order:
step 1:using input information bit vectors
Figure 458201DEST_PATH_IMAGE038
And a check matrix
Figure 639783DEST_PATH_IMAGE033
Multiplication direct calculation
Figure 658817DEST_PATH_IMAGE049
；
Step 2Using input information bit vectors
Figure 293061DEST_PATH_IMAGE038
And a check matrix
Figure 859171DEST_PATH_IMAGE035
Multiplication direct calculation
Figure 223157DEST_PATH_IMAGE050
；
And step 3:using intermediate result vectorsAnd input information bit vector in check matrix
Figure 33167DEST_PATH_IMAGE038
And a check matrix
Figure 402968DEST_PATH_IMAGE042
In
Figure 621460DEST_PATH_IMAGE013
Computing a codeword vector
Figure 418515DEST_PATH_IMAGE043
As follows:
Figure 27350DEST_PATH_IMAGE044
and 4, step 4:: combining the results of the step 1 and the step 3 to finally obtain the code word
Figure 499045DEST_PATH_IMAGE023
。
Example (c): the correction of the quasi-cyclic low-density parity check code and the linear coding method thereof can pass through
The following examples illustrate. An original check matrix of a quasi-cyclic LDPC code with code length of 6, information length of 2 and code rate of 1/3 and hidden nodes is as follows:
Figure 509727DEST_PATH_IMAGE051
。
the check matrix can be decomposed into 3 parts
Figure 540000DEST_PATH_IMAGE008
Wherein
Figure 636132DEST_PATH_IMAGE052
,
Figure 410053DEST_PATH_IMAGE053
,
Figure 275240DEST_PATH_IMAGE054
。
Figure 414098DEST_PATH_IMAGE009
information bit portion corresponding to the complete codeword of size
Figure 59843DEST_PATH_IMAGE010
In this example, not sent on the channel, and is thus a so-called hidden node part;
Figure 575138DEST_PATH_IMAGE011
corresponding to a parity bit portion of codeword dimension 1, of size
Figure 858614DEST_PATH_IMAGE012
；
Figure 168372DEST_PATH_IMAGE013
Corresponding to a parity bit portion of codeword dimension 2, of size
Figure 301413DEST_PATH_IMAGE014
(ii) a Parameter(s)
Figure 354820DEST_PATH_IMAGE055
,. Of the check matrixThe corresponding nodes form a large ring, as shown in fig. 1.
The structural correction of the invention is to select a point on the large ring to cut off one edge, namely to fill 0 in the '1' of the corresponding position of the check matrix, and to set the circled position of figure 1 as '0', thus the corrected position isCan be written as:
Figure 949432DEST_PATH_IMAGE058
，
thus, the check matrix of the LDPC code with the modified structure can be written as follows:
Figure 643719DEST_PATH_IMAGE059
。
the linear coding algorithm of the LDPC code after the structure modification comprises the following specific steps:
step 1:using input information bit vectorsAnd a check matrix
Figure 904378DEST_PATH_IMAGE061
Multiplication direct calculation
Figure 299587DEST_PATH_IMAGE062
；
Step 2Using input information bit vectors
Figure 176277DEST_PATH_IMAGE060
And a check matrixMultiplication direct calculation；
And step 3:using intermediate result vectors
Figure 526989DEST_PATH_IMAGE041
And a check matrix
Figure 258185DEST_PATH_IMAGE042
In
Figure 251549DEST_PATH_IMAGE013
Computing a codeword vector
Figure 5878DEST_PATH_IMAGE064
Calculated step by step as follows
Figure 318391DEST_PATH_IMAGE065
,
Figure 841776DEST_PATH_IMAGE066
)：
Figure 333937DEST_PATH_IMAGE067
,
Figure 575563DEST_PATH_IMAGE068
,
Figure 913003DEST_PATH_IMAGE069
,
Figure 556474DEST_PATH_IMAGE070
；
And 4, step 4:: combining the results of the step 1 and the step 3 to finally obtain the code word
Figure 891640DEST_PATH_IMAGE023
。
FIG. 1 is a large ring structure of a submatrix with variable dimension 2 in a quasi-cyclic LDPC code check matrix. The coordinates of the points in the ring in the matrix are
Figure 682879DEST_PATH_IMAGE071
Wherein
Figure 761693DEST_PATH_IMAGE065
,
Figure 823452DEST_PATH_IMAGE066
。

Claims (7)

1. A quasi-cyclic low-density parity-check code is characterized in that the dimension of variable nodes of the parity-check code is divided into 3 types: dimension 1, dimension 2, and dimension greater than 2; variable nodes with dimension larger than 2 are all information nodes, correspond to information bits to be coded, and are called hidden nodes because the variable nodes are not generally sent to a channel; the variable nodes with the dimension of 2 just form a large ring connected end to end on the bipartite graph corresponding to the corresponding low-density check matrix.
2. A method for correcting quasi-cyclic low density parity check codes according to claim 1, characterized in that the method comprises: optionally selecting one edge on the large ring with the dimension of 2, and cutting off the edge, namely filling 0 in the corresponding position of the low-density check matrix, thereby obtaining a modified structure of the code; and filling a row in which the 0 operation is positioned in the check matrix as a position for starting coding, and calling the row as a coding starting row.
3. The method of correcting a quasi-cyclic low density parity check code according to claim 2, wherein the correcting structure now modifies the check matrix of the low density parity check code, the modifying operation is to take the columns of which the column constituting the large ring is 2 in the check matrix, and to set "1" of any one column to "0", the row where the replacement occurs is called a start row, and the correcting structure is specifically expressed by combining the definition of the check matrix as follows:
defining: a class of check matrices with implicit node quasi-cyclic low density parity check codes:
Figure FDA0000040551030000011
wherein Hi，jIs a cyclic shift permutation sub-matrix of size zxz, which is completely dependent on the cyclic shift offset, mbPermuting the number of rows occupied by a sub-matrix for cyclic shifts in a check matrix, nbPermuting the number of columns occupied by the submatrix for cyclic shifts in the check matrix, HoThe matrix size is m × n ═ mbz×mbz; for ease of encoding, such check matrices are divided into 3 parts:
Ho＝[Hs|Hp1|Hp2]，
wherein HsThe information bit portion corresponding to the complete codeword, size m × k; single-dimensional check matrix Hp1To pair
-1-
Check bit part with size of m × n corresponding to code word single dimension column weight of 11(ii) a Two-dimensional check matrix Hp2A check bit part with a length of m × n corresponding to a two-dimensional column of 2 of a codeword2(ii) a The total length of the code is n ═ k + n1+n2(ii) a Due to HsThe information bit portion of the corresponding encoded codeword is not sent onto the channel and is thus a low density check code with implicit nodes.
4. The method of claim 3, wherein the bi-dimensional check matrix H is a parity check matrixp2All '1's form a large ring, and a two-dimensional check matrix H is arrangedp2The coordinates of the '1' in the matrix in the counterclockwise sequence of the macrocycle are sequentially
Figure FDA0000040551030000021
5. The method of claim 3, wherein the bi-dimensional check matrix H is a parity check matrixp2One of the '1' is arbitrarily established, the '0' is set, and the modified two-dimensional check matrix is recorded as
Figure FDA0000040551030000022
The final modified structure low density check code has a check matrix:
Figure FDA0000040551030000023
6. a method for linear coding of a quasi-cyclic low density parity check code according to claim 3, characterized in that: and calculating the coding bit by using the low-density check matrix of the modified structure and the input information bit vector: firstly, multiplying an input information vector s and a part of a check matrix with the column weight more than 2 by a vector and the matrix to obtain an intermediate vector u; coding vector with variable node dimension of 1
Figure FDA0000040551030000024
Directly intercepting the corresponding position of the intermediate vector u; coding vector with variable node dimension of 2Then the binary code vector is calculated bit by starting from the starting bit through the large ring characteristic of the binary code vector on the bipartite graph, and the two parts of code vectors are spliced to finally form the code output vector
Figure FDA0000040551030000026
7. The linear coding method of quasi-cyclic low density parity check code according to claim 6, wherein the coding vector is divided into two parts, one part corresponding to the column with the column weight of 1 of the check matrix is obtained by directly coding the information vector; the other part corresponds to the column with the check matrix column weight of 2, the corresponding coding vector can be obtained by bit-by-bit calculation through the large ring characteristic, and the coding algorithm is specifically expressed as a plurality of steps which are executed in the following sequence:
defining: let the input vector of the encoder be s ═ s1，s2，L，L，sk) Where k is kbz; the output of the encoder is a coded codeword, noted
Figure FDA0000040551030000027
If the information bit corresponds to the hidden node, the information bit is not transmitted and the output of the encoder is
Figure FDA0000040551030000028
Wherein,
Figure FDA0000040551030000029
a code vector corresponding to the one-dimensional check matrix, the size of which is set to n1：
Figure FDA0000040551030000031
A code vector corresponding to the two-dimensional check matrix and having a size of n2(ii) a Will matrix HsWrite to a partitioned matrix
Figure FDA0000040551030000032
Wherein Hs1Is n1×k，Hs2Has a size of (m-n)1) X k, and m-n1＝n2；
Step 1: using the input information bit vector s ═ s1，s2，L，sk]And a check matrix Hs1Multiplication direct calculation
Figure FDA0000040551030000033
Step 2: using the input information bit vector s ═ s1，s2，L，sk]And a check matrix Hs2Multiplication direct calculation
Figure FDA0000040551030000034
And step 3: using intermediate result vector u and input information bit vector s ═ s in check matrix1，s2，L，sk]And H in the check matrix Hp2Computing a codeword vector
Figure FDA0000040551030000035
As follows:
and 4, step 4: : combining the results of the step 1 and the step 3 to finally obtain the code word
Figure FDA0000040551030000037
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Application publication date: 20110511