JPH03172026A - Encoding/decoding system - Google Patents

Abstract

PURPOSE:To enlarge an encoding rate (information length/code length) by correcting plural symbol burst errors with the addition of efficiency redundancy. CONSTITUTION:Input data (u) are stored in a data buffer 8 and simultaneously, a vector V1 is calculated while multiplying a first matrix by a constant multiplying circuit 2. Next, an eraser position selecting circuit 3 designates a first eraser position as eraser information and the information are sent to a decoding circuit 4 and decoded as syndrome in a part of the V1 by a first decoding means. Afterwards, an eraser position selecting circuit 5 designates a second eraser position as the eraser information and sends the information to decoding circuit 6 and the residual part of the V1 is defined as a second syndrome and decoded by a second decoding means based on the previously designated eraser information. Then, a constant matrix multiplying circuit 7 multiplies the second matrix to a symbol train V2 and a symbol train V3. Consequently, the symbol train, which is subtracted from the input data (u) by a vector adder circuit 9, is outputted from an output terminal 10.

Description

Detailed Description of the Invention (Industrial Application Field) The present invention provides a code for automatically correcting symbol errors received on a communication channel or storage medium on a receiving side in a digital communication system or digital storage system. Regarding the decoding north method.

(Prior Art) It is known that errors caused by the transmission or storage of digital data are often caused by noise on the transmission path or physical defects in the storage medium. Among these, practical codes that can deal with multiple symbol burst errors have recently attracted particular attention, and in magneto-optical disk devices, interleaved Reed-Solomon codes (IRC) and Reed-Solomon codes that double combine Reed-Solomon codes are used. A product code (RPC) or the like is adopted. These codes are also used as codes that simultaneously correct random bit errors as well as bit burst errors. In general, existing codes (for example, Reed-Solomon codes, which are maximum distance separation (MDS) codes, etc.) are multiplexed and interleaved,
A so-called product code, which performs multiple (for example, double) combinational coding of existing codes, is known. 19 from The MIT Press in the United States
W. W. Peterson and E. J. Weldon, J., published in 1972.
``Error Correcting Code (Erro
r-CorrectingCodes) J 357-
On page 359, the interleaving method is explained limited to cyclic codes, and on pages 131-136 there is an explanation of product codes. In addition, 19
``Key Points of Error Techniques 1'' published in 1986, supervised by Hideki Imai, contains information on product codes on pages 36-37, and articles 39-40.
There is a description of the interleaving method on page 1 and page 215.

Also, IEEE Transactions on Information Theory (IEEETRANSA)
C-TIONS ON INFORMATION TH
281 of Volume IT-11 (1965) of EORY) magazine
- J.K. Wolf on page 284
On Codes Derivable from the Tensor Product of Check Matrix (On Codes Derivable from the Tensor Product of Check Matrix)
m the Tenso Product of'Ch
eck Matrices) J, and the IEEE Transactions on Information Theory (IEEE Transactions on Information Theory) which further expands this.
CTIONSONINFORMATION THEOR
IT-27, No. 2 (1981), No. 18 of Y) Magazine
HIDEKI IMA on pages 1-187
I), HIROSHI FUJITA
)'s paper on Generalized Tensor Product Code (Generalized Tensor Product Code)
There is research on a series of codes called tensor product codes, such as in Product Codes. However, this is only research on codes aimed at correcting random bit errors, and the codes targeted are multi-dimensional codes and two-way codes.
It is a binary code that combines the original code. On the other hand, the MDS code type tensor product code according to the present invention is modified to be suitable for correcting multiple symbol burst errors. In other words, it proposes a tensor product code as a general multidimensional code that uses a combination of a multidimensional code and a multidimensional code instead of a binary code (by modifying it, it has the structure as a white multidimensional code), and furthermore, Among such configurations, the code structure is limited so as to be more adaptable to multiple symbol burst errors.

Now, the 109th error correcting code mentioned above.
According to page 110, in general, in a linear block code where the number of redundant symbols is r, a (symbol) burst of arbitrary length j can be corrected, and a (symbol) burst of arbitrary length i (i≧j) can be detected. The necessary and sufficient condition for
≧i+j” holds true. Therefore, as a code for correcting symbol burst errors, it is desirable to achieve the limit r=i+j of this general limit equation. The interleaved evil code mentioned earlier achieves this limit (the decoding method has also been established). Therefore, i=
By setting j, symbol burst errors of less than half the number of redundant symbols can be completely corrected. Furthermore, even if a plurality of symbol burst errors occur, they can almost be corrected if the sum of the symbol burst lengths is less than half the number of redundant symbols (not in a strict sense).

(Problem to be Solved by the Invention) However, on the other hand, the interleaved RS code has a drawback that as soon as the sum of symbol burst lengths exceeds half of the number of redundant symbols, most of them cannot be corrected. In order to solve this problem, the present invention proposes an error-first method that (effectively) corrects multiple symbol burst errors with less redundancy, and provides its encoding and decoding method.

In the proposed code, some exceptional error patterns whose sum of lengths is within half the number of redundant symbols and which are not correctly corrected will occur, but these can be kept to a sufficiently low level, and the length of symbol burst errors can be reduced. It is possible to correctly correct most items up to the sum of just a few redundant symbols. It should be noted that a product code or the like can also be used to create a code that has a similar effect, but it should be noted that it is far inferior in terms of redundancy efficiency.

(Means for solving the problems) In order to solve the problems described above, the first invention of the present application has the following features:
In an encoding method that performs error-first encoding to correct a plurality of symbol burst errors occurring in data, the method includes means for accumulating input data U (vector = symbol string), and A part of the vector (syndrome symbol string) Vl multiplied by a matrix of 1 is set as the first syndrome, and the erasure position is set to 0. 1
, 2, . 1, 2, . Based on the method of outputting a symbol string obtained by subtracting the multiplication result (symbol string) from the input data U stored in the storage means, matrices Hd-1, Hd-, Let L,, be false first check matrices with the property of maximum distance separation, and the sizes are (d-1)×n, s×n, and g×h, respectively,
Let matrix Ih − g, h have size (h − g) × h,
Assuming that a square matrix of size h×h is regular, the size of the first matrix H when the symbol O is the tensor product of the matrices is ((d-1)-g+(h- g)s)
×n h matrix on F, and the input data U is u
u1,. l u1, 1+ '..."1, 1" 2,.

, u2, 1'"" u2, 1' uh, 0' u
h,! '"" uh, 1) length uj, m(1≦j
≦g, o≦m≦d−2) and u. (g+1≦j≦h,
o≦m≦J, mJ, m syndrome (Hd
4■L,,). u, and the first decoding means is the matrix H
A decoding means executes the first error decoding with d-1 as the check matrix g times, and the second decoding means executes the first error decoding with the matrix H as the check matrix g times. The present invention is characterized in that the decoding means executes decoding h−g times.

Further, the second invention of the present application provides a decoding method for performing first error decoding for correcting a plurality of symbol burst errors occurring in data, in which input data U (vector=
A part of the vector (syndrome symbol string) Vl obtained by multiplying the input data U by a first matrix is decoded by a first decoding means as a first syndrome. vector (symbol string)
is set as V2, then the erasure position is set as the error position regarding the V2 detected by the execution of the first decoding means, and the remainder of the hector V1 is set as the second syndrome, and then decoded by the second decoding means. A symbol string is output by subtracting the result (symbol string) of multiplying the decoded vector (symbol string) V3 and the second matrix by the second matrix from the input data U stored in the storage means. Based on the method, the matrices Hd-, , Hd-, Lg, and h on the finite field F are defined as error-first check matrices having the property of maximum distance separation, and the sizes are (d-1)×n and s×, respectively. n, g×h, and the matrix Ih−g,h has the size (h−g×h, and a square matrix of size h×h is regular.) The size of the matrix H is ((d-1
)−g+(h−g)・s)×n・h matrix on F, and the input data U is encoded by the encoding according to claim (1). The possible received vectors u,. pu1.1p..., u1,
1' The decoding means is a decoding means that executes the first error decoding using the matrix Hd4 as a check matrix g times, the second matrix is a matrix, and the second decoding means is the matrix H. The present invention is characterized in that the decoding means executes the first error decoding h−g times using the parity check matrix as the check matrix.

(Embodiment) FIG. 1 is a block diagram showing an example of circuit implementation for realizing the encoding/decoding method according to the present invention. Note that encoding and decoding are realized by circuits with almost the same structure. The only difference between the encoding side and the decoding side is the selection of erasure positions.

The input data U (vector 1 = symbol string) of the present invention is input to the input terminal 1 and stored in the data vanofer 8, and at the same time, a constant matrix is multiplied by the multiplication circuit 2 via the line 11 to the input data U. A vector (syndrome symbol string) Vl is calculated by multiplying the matrix. Next, the erasure position selection circuit 3 specifies the first erasure position as erasure information and sends it to the decoding circuit 4 (which executes the first decoding means), where the same line 12. The first decoding means decodes the syndrome as part of v1 via V13. The decoded vector) (symbol string) is V2
shall be. Next, the second erasure position sent via line 16 is designated as erasure information by the erasure position selection circuit 5 and sent to the decoding circuit 6 (which executes the second decoding means). The remainder of the vector V1 sent via the second syndrome is decoded by the second decoding means based on the erasure information specified above. The decoded vector (symbol string) is assumed to be V3. Next, in the constant matrix multiplication circuit 7,
Symbol string ■2 sent via line 14 and line 1
Multiply the symbol string 3 sent through 8 by the second matrix. The result (symbol string) is sent to the vector addition circuit 9 via the line 15, and the input data U sent from the data buffer 8 is sent to the vector addition circuit 9.
The symbol string that has been drawn down is outputted from the output terminal 1o via the line 22.

However, the matrices Hd-1, Hd-L on the finite field F. h
Let Ih-g and h be false-first check matrices with the property of maximum distance separation and have sizes (d-1)×n, s×n, and g×h, respectively, and let matrices Ih-g and h have sizes (h- g)×h, and the square matrix of size h×h is regular, then the first matrix H has a size ((d−
1) −g+(h g)−s)×n·h matrix on F. In addition, when input data U is encoded, u is changed to u1, 0' ul, 1'"" uI, 1' u2, 0
' u2, 1' ”” u2, 1' uh, O' uh
, 1+ "" uh, u in 1). (1≦j≦
g+O≦m≦d−2 and U. J + m
J, m(g+1≦j≦
h, o≦m≦s−1) total (d−1)・g+s−(h−
g) Set the symbol to zero as a redundant symbol and create a vector with the rest as information symbols. Also, when decrypting 1
Ma” ” (u1, 0' u1. 1' ”” u1,
1' u2, 0' u2, 1'"" u2, 1'u
h, O"h, ]'"" uh n-1) is a received vector in which an error may have occurred.The first erasure position is 0, 1, 2, . . . during encoding.
, d-2, and are assumed to be absent at the time of decoding. The first syndrome is (Hd--1Lg,h).U. The first decoding means means a decoding means that performs the first error decoding g times using the matrix Hd-1 as the check matrix. What is the second matrix? The second erasure position is 0 when encoding
, 1, 2, . ．．・,s-2, and when decoding,
Let it be the error position regarding V2 detected by the execution of the first decoding means. The second syndrome (Hd-■Lh-
g, h'・U. The second decoding means means a decoding means that executes the first error decoding h−g times using the matrix H as a check matrix.

This will be explained in more detail below. The relationship between FIG. 1 and the encoding/decoding method according to the present invention will become clearer in "Relationship between FIG. 1 and the decoding procedure" and "Relationship between FIG. 1 and the encoding procedure", which will be described later.

Here, an MDS code (maximum distance separation code) type tensor product code is defined on the finite field F=GF(q), and its encoding/decoding method is presented. However, for the sake of explanation, M
A typical example is the DS code. Ree
The discussion will be limited to d-Solomon codes (hereinafter referred to as evil codes). In addition, in the following, it is normally defined on F. [n, n - d + 1, d] Assuming that the decoding method of e-erasure/7 random symbol error correction of R8 code (having q types of symbols) is a well-known fact (i.e., known), the decoding circuit is decoded. RS-D
It is expressed as ECODER. It is the syndrome 3*,, 3*, which is calculated from the reception bee, L.
..., S book, -2 and erasure position as input,
This is a decoding circuit that outputs a transmission vector (code word) and an error vector expected as a result of decoding, or an error detection signal indicating that correction is impossible. However, d≧2t
+e+1. For details, see the book ``Error・
Correcting Code 1, pages 305-307 or Error-Correct Coding for Digital Communications by Clark Cain.
ion Coding for Digital Com
It is written on pages 214-225 of a book published by PLENUM PRESS in 1981 called ``Munications'' J.

A. Sign parameter of code to be realized Code parameter of code to be realized here nRSTIkRST? dRS
T [code length, information length, minimum distance] is [nRST "h"1kr? when designing natural numbers S and d as 8 = [(d - 1)/2]. Sr=h−n−(d−
D-g-(h-g)・s, dRs. ,,≧d]. As an absolute guideline, S random error or ([
s/2] -1) h+g+1 symbol burst error first.

However, as will be discussed in detail later as an evaluation of the correction ability of r Errors are also codes that can be correctly corrected.

B. Definition of RS-type tensor product code Based on the code on F, the residence-type tensor product code RST is defined as follows.

a(F is a primitive element, matrix H. (j=s, d-1),
L. ,,'' Here, the matrix H. We have presented a parity check matrix for a certain evil code, but below we will discuss H. ,
Or H. The matrix that has been commonly shortened for each j (=s, d-1) (by deleting some columns) is rewritten in H. Let the number of columns J be n (t<n≦q+1). Note that the first two columns of the H. is the check matrix of the cyclic evil code.

J Based on these, ((d-1)・g+(h-g)・on F
s) Define the X(n-h) matrix H.

More specifically, column H can be written as follows.

H= Also, when g is set to O, the so-called internove
This is the check matrix itself of the Reed-Solomon code. The code that uses the matrix H defined here as a check matrix is called Reed-Sol
The omon type tensor product code is referred to as RST. As noted at the beginning, the matrix Hd−−,, H
Any check matrix of the MDS code may be used as d-, L, and h. Also, (h-g)×h
The matrix Ih − g, h may be any matrix as long as the h×h matrix is regular. Here, most simply 'h-g,
I chose h. Also, from this point forward, the parameters s and d are [
(d − 1)/2] = s is assumed to be satisfied.

C. Explanation of terms for decoding (d-1) Let the Xn matrix Hd-4 be the check matrix [n, n
+1-d,d]RS code as RS. (1≦j≦g), and check the sXn matrix H. Let the matrix be [n, n-s, s+ IIRS code as code RS. It is called (g+1≦j”≦h).

The transmission vector is “”■1,0,■1.1'””v1,
1, ■2.01 ■2.11””v2, n-1”h
, 0"h, 1'"" vh, 1) [F]
This is the reception vector.

Also, a parity check matrix H is created for the received vector U. Here, for the code RSi, (1≦j≦g), M, the transmission vector, M. Reception vector, M, called error vector j
h(if)・i−1 h(i−1)i−1■
．． *Mi (Σα V. Σα t.1,...'
i=1 ”・” i=1h (i-1)・
i-1 Σα v, 1,) (F, i=1 h (i-1)i-1 h (j-1)・i
-1U, et al (Σα U, Σα ui, 1'...
J; :t l,0ko=1?41"1゛・1,.1■)(F゜, Σα i=1 h(jl)・i−1h(i−1>・i−1e.Nemi( Σ
α e. Σ α' i=1
'・” i=1 ”・l′“゜-“
゜"−゛・1,.-1,)(F", defined as Σα i=1 (depends on matrix L.,), and the sign RS, (
Similarly, for g+1≦j≦h), b transmission vector,
J, which is called b reception vector and M error vector, are respectively expressed as Vj*1(VJ.or Vj.1+ ”゜+Vj.n
-1)) (FnNuj*a(uj, o, uj, 1, -,
u, 1)) CF”, ej*mi(ejIo, ej1,...
・, ejln-1)) (Define as Fn1 (matrix Ih-
g, h), then U, books = v. ” + e
．． ”, then v41, code RS. (1≦j≦h) as J.J J
It becomes J-word. Furthermore, the relationship between each M, the reception vector, and the syndrome is Hd-1゜uj''= (S(d-1)(j-1)l S
(d-1)(I-1)+11..."l S(d-1)(
j-1), 1-2)+(1≦j≦g), S(d-11.g+s.(j-g1.ya,-2), (g
+1≦j≦h).

D. Syndrome S for received vector U at the beginning of the decoding procedure. ,S1
,...' S(d-11.g+s-(h-g)-1 is derived from the above relational expression. Next, for each j (1≦j≦g), the syndrome S(d-1), −1)+i(
(or,
If the erasure position regarding the column vector in the matrix representation of the transmission vector is presented by some prior information, the decoding RS-DECODER is performed in consideration of the erasure position.
Execute t. Regarding this point, please refer to ``Evaluation of correction ability of rRsT code'' below. However, in this case, set S in advance so that s=[(d-1)/2]<[(d-
1)/2]<s≦d). Here, j (1≦j≦g
) cannot be corrected, the code RS is added at this point.
Even when decoding T, a signal indicating that correction is impossible is output, and the decoding ends. Also, all j (1≦j≦g)
The estimated M. Error vector J vector eJ*EJ+o, EJ+"...' Ej,
Let m'..., Ej, 1). Here, natural numbers e and t are defined as follows. e=#{m: some j (1≦j≦g
) regarding E. ≠0, but 0≦m≦n-1}J,m Also, let t be the largest natural number that satisfies s+1≧2t+e+1. If e≧s+1 at this time, a signal indicating that correction is impossible even when decoding the code RST is output at this point, and the decoding is terminated. Also, when S≧e,
For each j (g+1≦j≦h), set the erasure position {m: for some j (1≦j≦g) E.
J + m ≠ 0, but all of m included in 0 ≦ m ≦ n-1}, and decoding RS-DECODER (of code RS) as e erasure/7 random error correction, syndrome J e,
LmuS(d-1)5g+s,-g-1)+i(1≦i≦
S) is executed as input. Here, j(g+1≦j
≦h), if it becomes impossible to correct, at this point the code R
Even when decoding ST, a signal indicating that correction is impossible is output and the decoding ends. Also, if all of these can be corrected, the estimated M. Error vector (g+1≦
j≦h), respectively.
-,E,. Let n-, ). At this time M. From the error vector (1≦j≦h) we can determine the expected error vector e as follows. That is, for any j (1≦j≦h), E. J
, m = 0 for m, e. =0 (1≦j≦h), then there is a certain j (1≦j≦h) such that E.≠0.
Regarding J,m, the error symbol e. (1≦j≦h) is determined J,m. In particular, for j (g+1≦j≦h), e. =
E. (0≦mJ, mJ, m≦n-1) (depends on matrix Ih − g, h).

This completes decoding.

Here, if we check again the case where the code RST becomes uncorrectable, we can see that a certain code RS. (1≦j≦g)” becomes uncorrectable, or the set of erasure positions {m: E,≠0 for some j (1≦j≦g), only J+m and O≦m≦n -1} truly exceeds S, or if the number of symbols RS. Either the decoding of (g+1≦j≦h) becomes impossible to correct.

E. The relationship between FIG. 1 and the decoding procedure is "input data U input to input terminal 1". "
The definite matrix in the definite matrix multiplication circuit 2 means the check matrix H. Therefore, ``vector VIJ is a syndrome vector (SO'S1'``'' S(d-11.g
+s-gl-,). For more details, V1=H−u
It is. Next, specify the first erasure position as erasure information in the erasure position selection circuit 3 and
In this case, it is assumed that a certain erasure position has not been specified. "Sent via lines l2, l3 t (S(d-x).(i-1,,s(d-1).(j-1
)+11 ””l S(d-1)(j-1)+d-2)
(l≦j≦g). "F decoding circuit 4" is coded RS
．． (1≦j≦g, note) Note: This is avoided by the decoding circuit RS-DECORDER, which performs normal decoding (common regardless of j). (The number of attention two-input syndromes is d-1). Here, as specified by erasure position selection circuit 3 (e, t
) = (0,S), and the syndrome S(d-11(j
-t)+s(d-1). (jo+11...S(d
−11(i−11+d 2 is input and decoding is j(1≦
This is performed g times depending on j≦g). Here, the estimated g number of M error vectors is E.

, Ej, 1, -...' Ej, m'...'' A vector consisting of Ej, 1' and a signal indicating whether or not it is uncorrectable in g decodings (all of these are considered as one vector) "means vector V2J.

That is, what outputs this is the decoding circuit 4J.

``The erasure position selection circuit 5 uses the second erasure position sent via line l6 as erasure information.
An erasure position specified by "g M. error vector e. *Garadoga J
Erasure position {m: Dj, nl≠O, where O≦m≦n−1} for some j (1≦j≦g). Let natural numbers e and t be expressed as e=ll(m: some j(1≦j≦
Regarding g) E. ≠0, just J,m and O≦m≦n-1}, and t=[(s-e)/2].

However, [-] is a Gauss symbol. ``The decoding circuit 61 consists of a decoding circuit RS-DECORDER, which performs normal decoding of the code RS.(g+1≦j''≦h. Note: common regardless of j). (
Note: The number of human cash syndromes is S). ``The remainder J of the vector 1 via line 15 means the syndrome g-1) + s-1) (g+work≦jsh).Here, the erasure position is specified as specified by the erasure position selection circuit 5. and each j (g+1≦j≦h
) for (e, t) = (e, t), and the syndrome s(d-11.g4-s(i-g-1)+1"
・, Sd-1). g+5(j-g-1)+s-1)(g
+”≦j≦h) is decoded h−g times. Here, h−g M. error vectors estimated here are ej = (Ej, o, Ej, 1t . . -, E
j, m'. ．． ．． , Ej, J. -1) A vector consisting of (g+1≦j≦h) and a signal indicating whether or not correction is impossible in g decodings (considered as one vector) means “vector V3J. In other words, what outputs this This is "decoding circuit 6."

The constant matrix in "Definite matrix multiplication circuit 7" means a matrix. "e." which is the component of ■2 and ■3 is E. E.J.
J. The error symbol e. (1≦j≦h) is determined from 0'J+"..., Ej, m, -..., E, 1XIO≦j≦h). Here, the symbols indicating uncorrectable judgment and (V2 and V
Judging by the uncorrectable symbols of 3: If either one is uncorrectable, it is determined that it is uncorrectable), and this expected error vector e = (e, .+ e1, 1+...,
e1,. -11e2,. le2,1+""e2,1'
It is the "definite matrix multiplication circuit 7" that outputs eh, O'eh,1'"" eh, 1).The error vector e just obtained is subtracted from the received vector U stored in the F data vanofer 8. The transmission vector v expected to be
=ue is obtained. This is executed by the "vector addition circuit 9". However, when the symbol indicating the uncorrectable judgment sent together with the error vector indicates uncorrectable, the received vector is outputted as is from the output terminal 10 along with the signal indicating uncorrectable.

F. Explanation of terms for encoding aj, m=0 (1≦j≦g, O≦m≦d−2), aj,
Let m=O (g+1≦j≦h, o≦m≦s−1), and for other j and m, a. J, m 2 v. shall be. Also b. =v. (1≦j≦g,
o≦m≦d−2), J, m
J,m J,mb. =v. (g+1≦j≦
h, o≦m≦s−1), and other J, m J
, m for j, m, b. =0. In other words, the transmission vector Jam tonore""5vl, Oj v1, II゜""1, 1
'2.0'2.1" 2, 1' h, 0'
vh, 1””, vh, 1), vj, (1≦j≦g
yo≦m≦d−2) and vj,m(g+1≦j≦h, o≦
A total of (d-1)-g+s (h-g) symbols where m≦s-1) are used as redundant symbols.

Ru.

Here the code RS. For (1≦j≦g), the transmission vector, M, the information vector, and M, the redundant vector are respectively expressed as h (i-])・i-1 ■$2(Σα J i=l IQI h (j-1 冫・i-1 Σa v ... i=1 '・l'' (i-1) it Σα vi,.) (F, i=1 h ri-1) i- 1 h (j-1)・i-1
a. Σα a Σq J, E1 '+O' 1:1 1
．． 1'h reel)・i-1 Σa a,. one. ) CF, i=1 h (j-1)i-1 h (j-1)i-
1b,′* as Σa b. Σa b.
...' i=1 '・0' i=
1 +, 1'h(i-1)i-t Σa b. ,. −2)) (defined by F1 i=1, (depends on the matrix L,,), and the sign RS,
Similarly, for (g+1≦j≦h), M, the transmission vector, M, the information vector, and M, the redundant vector, are respectively expressed as vj* by vj,o, vj,1,−, v,■)) (F
, aj” to aj,o, a,1,−, aj,1,)
(F ,bj*j(bj,0・bj,l・゜゜゜・bj
, -1)) (Defined by F (matrix Ih − g, h
), then V, *=a book + b. ne next v
．． * is the code RS. Code JJ as (1≦j≦h)
J It becomes J J language.

It becomes a word.

G. Encoding procedure Encoding the code RST is equivalent to determining a unique redundancy vector for an arbitrarily given information vector.

Syndrome S is first derived from parity check matrix H for the information vector. ,S1,...,s,-2,...S(
d-1). Find g+g-(h-gl-1). Next, set the erasure position to 0 for each j (1≦j≦g).
, 1, ..., d-2 (with code RS.)
DECODERdt,a syndrome' S(d-1). (i-.+, (0≦i≦d-2) is executed as input. Also, for each (g+1≦j≦h), the erasure position is set to 0.1, -, s-1 (sign RS.) Decoding RS-'DECODER,o as syndrome S(d-1)g+s
, -g-1)+i (0≦i≦s-1) as input. Based on these results, M. Let the error vector be -b:- (Ej.o, Ej, 1,...J
'E,,m,-
, E, n, XI≦j≦h) and vertical redundant vector bj
Derive * (when the characteristic of the field is 2, - is not needed). At this time b. * (1≦j≦h), determine the redundant vector J as follows. That is, the redundant vector b has each m (0≦
m≦d−2) by the following relational expression b. =v
．． (1≦j≦g, o≦m≦d−2), b. =v
．． ',m J,m
Jam J+m (g+1≦j≦h
, o≦m≦8-1). Encoding is now complete.

H. Relationship between Figure 1 and encoding procedure ``Input data uJ input to input terminal 1 means information vector a in this case a1.Pal.ll...,
a1,. 4, a2, 0' a2, 1' ”” a2,
n-1'"" ah, 0' ah, 1'""
ah, n-1) ' aj, m=0 (Work≦j≦g, 0
≦m≦d−2), aj, m=0(g+I≦j≦h,
O≦m≦$−1). The constant matrix in "definite matrix multiplication circuit 2" means the check matrix H. Therefore, "vector VIJ is a syndrome vector" (S
(d-1). (i-11' S(d-11-(i-.+
1,...l ””(d-1)(i-1)+d-2”≦
j≦g). In this case, the erasure position that says ``Specify the first erasure position as erasure information in erasure position selection circuit 3'' is 0, 1,
2,...,d-1. "Decoding circuit 4" is coded R
S. A decoding circuit RS-DECORDER that performs normal decoding of J (1≦j≦g. Note: common regardless of j) is used. (Note e, L, the number of two-person syndromes is d-1). Here, each j
For (1≦j≦g), (e, t) = (d-1.
0), the syndrome Hd-1' aj”
” (S(d-4)(i-11”(d-1)5(j-1
)+1'...' S(d-1)(j-1)+d-2)
(This is the "first syndrome that is part of ■1") as input and is repeated g times. The g M. Let the error vector be j −b. Mon:- (E,., E,,1,...'l Ej
, mt ``'l Et,n l) to obtain a redundant vector J vector b*, but the vector that combines these vectors'' (considered as one vector) means ``vector V2J. In other words, this is output The one that does this is “decoding circuit 4”
It is. ``The second erasure position sent via line 16 is designated as erasure information by the erasure position selection circuit 5, and the erasure position 1 is unchanged in this case and is 0, 1, 2, ..., s-. Set to 1.

"Decoding circuit 6" has the symbol RS. (g + 1 ≦ j ≦ h. Note: Note: The number of ROMs is S. ). ``The rest of the vector node 1 via line 15'' means to send the book to syndrome Hd-1 S(d-1).g+s.(i-g-1)+1'...
・' Sd-1). g+s. (i-g-1)+s-1)
(g+1≦j≦h). Here, each j (g+1≦j≦h) as specified in the erasure position selection circuit.
The decoding with (e, t) = (s, 0) for the syndrome S(d-1)g-1-s(i-g-1) +
1' ”” S (d-11g+go-g-1)+s-1
)('''≦j≦h) is input and the process is repeated h-g times.The estimated h-g M error vectors are expressed as -b.ne:-(Ej, o, Ej, 1 ,...J
JEj, m'・
..., Ej, 1), M, redundant vector bj* is obtained, but a vector consisting of these (considered as one vector)
"means vector V3J. That is, what outputs this is the "decoding circuit 6." “Definite matrix multiplication circuit 3
"Definite matrix" means a matrix. bj* = (E
,,. + E;. Is...,E,,m,...,E
, ,n-XXi≦j≦h), the redundant symbols are b. =v. (1≦j≦g, 0≦m≦J+”
J+" d-2), b. = v. (g+1≦j≦h, o≦m
ss-1) J,m is determined as J,m. Encoding is now complete.

? ．． Evaluation of the correction ability of code RSTff The symbol burst error handled by this code RST is determined by changing the transmission vector v to v1. ,v1,■,...”1,1”2
．． 0"2. 1'...v2, 1'""l vh, O
``h, 1'''・, vh, 1)'' (bl, O' b,
1'"" a1, 1'b2.0' b2. 1'
・・” a2, 1) ”・, bh, 0' bh, lj
ah, 1), Vt, 1'' 2, 1'.'
1, 1””’Vh, 1”””2, Pv1, 1”h, 0”
”V2.., v1,.).

In other words, to explain in more detail in terms of a transmission vector matrix, it starts from the lowest symbol in the nth column on the right and moves upwards, then from n-1
It is assumed that symbol burst errors occur from the bottom of the column to the top, and successively in each column from the bottom to the top, all the way to the leftmost uppermost symbol.

Obviously, in this case, by decoding the code RST, any [
s/2] column can be completely corrected. In particular, for single symbol burst errors (I
s/2]-1, h+g+1 symbol burst error first. Furthermore, even if the number of columns in which errors occur exceeds [s/2], most of the error patterns up to column S can be correctly corrected.

This is because an error occurring in the S column will ultimately be incorrectly corrected or become uncorrectable in the code RST only in the following cases. That is, the code RS. In the decoding of (1≦j≦g), the erasure position information is missed,
Furthermore, due to the lack of information caused by this, the code RS
．． (g+1≦j≦h) is also corrected. Here, we will give necessary and sufficient conditions that such an exceptional error pattern should satisfy. First, the target error pattern consists of exactly r columns (
It is assumed that an error occurs in 0≦r≦S). In this case, the necessary and sufficient condition for an arbitrarily given error pattern to be such an exceptional one is that r
All the error vectors in a certain u (r≧U≧s−r+1) column among the columns are the code words of the [h, h−g, g+1]Rs code on F whose code length is h and the minimum distance is g+1. First of all, it is necessary that
It is also required that the value be 1 or more. From this result, these numbers can be understood quantitatively, which is very convenient for determining g in code design. It can also be confirmed from this that the number of these error patterns is extremely small.

(Effects of the Invention) As described above, since multiple symbol burst errors can be corrected by adding efficient redundancy,
Compared to conventional codes, the coding rate (information length l code length) can be significantly increased by ensuring the same error rate after correction. Further, the circuit scale for realizing the encoding/decoding method is not much different from that of conventional methods (for example, IRC, RPC).

[Brief explanation of the drawing]

FIG. 1 is a block diagram showing one embodiment of the present invention. In the figure, 2, 7... constant matrix multiplication circuit, 3, 5...
... erasure position selection circuit, 4, 6 ... decoding circuit, 8 ... data buffer, 9 ... vector addition circuit.

Claims (2)

[Claims] (1) In an encoding method that performs encoding of an error correction code to correct multiple symbol burst errors that occur in data, input data u (vector = symbol string)
, a part of the vector (syndrome symbol string) V1 obtained by multiplying the input data u by the first matrix is set as the first syndrome, and the erasure position is 0, 1, 2, . . . d-2 is decoded by the first decoding means, the decoded vector (symbol string) is V2, and then the erasure position is 0, 1, 2, ..., s.
-1, the rest of the vector V1 is decoded by the second decoding means as a second syndrome, and the result (symbol string) of multiplying the decoded vector (symbol string) V3 and the second matrix by the second matrix is stored. Based on the method of outputting the symbol string subtracted from the input data u stored in the means for
_h is a check matrix of an error correction code having the property of maximum distance separation, and the sizes are (d-1)×n, s×n, and g×, respectively.
h, and the matrix I_h_-_g_, _h has a size (h-g)×h, and the square matrix of size h×h [L_g_,_h I_h_-_g_, _h] is regular. When, the size of the first matrix H is ((d-1)・g+(h-g)・s)×n when the symbol ■ is the tensor product of matrices.
・In the matrix of h on F, set H■[H_d_-_1■L_g_,_h H_s■I_h_-_g_,_h], and set the input data u to u=(u_1_,_0, u_1_,_
1,..., u_1_, _n_-_1, u_2_, _0
, u_2_, _1, ..., u_2_, _n_-_1,
u_h_, _0, u_h_, _1, ..., u_h_,
u_j_, _m(1≦j≦g,
0≦m≦d−2) and u_j_,_m(g+1≦j≦h,
Total (d-1)・g+s・(h-g) of 0≦m≦s-1)
The first syndrome is defined as (H_d_−_1■L_g_,
_h)・u, and the first decoding means is the matrix H_d_
−_1 is a decoding means that decodes an error correction code g times as a parity check matrix, the second matrix is “L_g_,_h I_h_−_g_,_h]＾−＾1, and the second syndrome is ( H_s■L_h_-_g_,
_h) ^t・u, and the second decoding means is the matrix H_
An encoding method characterized in that a decoding means decodes an error correction code using s as a parity check matrix h−g times. (2) A decoding method that performs decoding of an error correction code for correcting multiple symbol burst errors occurring in data, which includes means for accumulating input data u (vector = symbol string), and the input data u A part of the vector (syndrome symbol string) V1 obtained by multiplying by the first matrix is decoded by the first decoding means as the first syndrome, and the decoded vector (symbol string) is set as V2, and then the erasure position is set as the corresponding vector (syndrome symbol string). Let be the error position regarding the V2 detected by the execution of the first decoding means, and the vector V
The remainder of 1 is decoded by the second decoding means as the second syndrome, and the decoded vector (symbol string) V3 and the V
2 multiplied by a second matrix (symbol string) is subtracted from the input data u stored in the storage means, and the symbol string is output. H_d_-_1, H_s, L_g_,
_h is a check matrix of an error correction code having the property of maximum distance separation, and the sizes are (d-1)×n, s×n, and g×, respectively.
h, and the matrix I_h_-_g_, _h has a size (h-g)×h, and the square matrix of size h×h [L_g_,_h I_h_-_g_, _h] is regular. When, the size of the first matrix H is ((d-1)・g+(h-g)・s)×n when the symbol ■ is the tensor product of matrices.
- A matrix of h on F, H■[H_d_-_1■L_g_,_h H_s■I_h_-_g_,_h], and the input data u is a symbol string encoded by the encoding described in claim (1). The received vector u = (u_1_, _0, u_1_, _1, ・
..., u_1_, _n_-_1, u_2_, _0, u_
2_, _1, ..., u_2_, _n_-_1, u_h
＿, _0, u_h_, _1, ..., u_h_, _n_
-_1), and the first syndrome is (H_d_-_
1■L_g_,_h)^t・u, the first decoding means is a decoding means for decoding an error correction code using the matrix H_d_-_1 as a parity check matrix g times, and the second matrix is [L_g_,_h I_h_-_g_,_h]^-^1, and the second syndrome is (H_s■L_h_-_g_,
_h) ^t・u, and the second decoding means is the matrix H_
A decoding method characterized in that a decoding means decodes an error correction code using s as a parity check matrix h−g times.