JPH01158826A - Error correction system for long distance code - Google Patents

Abstract

PURPOSE:To decode a long distance code at a high speed and to obtain an error correction system suitable for the hardware by applying the left basic transformation to a prescribed matrix so as to transform it into an upper triangle matrix thereby obtaining a coefficient of an error location polynomial. CONSTITUTION:In the decoding of a long distance code, a data word A(i+j-2) [1<=i<=p, 1<=j<=p+1, A0-A2p-1 are syndrome or error location] is set to each element q1, j of a matrix in p-row and (p+1) column to obtain a coefficient of each term or an error pattern of an error location polynomial. Then the left basic transformation is applied to the matrix to transform it into an upper triangle matrix thereby obtaining the coefficient of the error location polynomial. Since number of times of arithmetic operations is reduced, the processing is quickened without imposing excess load to an arithmetic unit.

Description

[Detailed description of the invention]

[Technical field] Encoding and decoding of data that uses error correction codes such as BCH codes and Reed-Solomon codes to correct errors in a large number of words, such as 255 words, as a block, on optical discs, etc. It is used for recording/playback and data transmission. An object of the present invention is to provide an error correction method for decoding such a code (long distance code) capable of correcting a large number of errors with one correction sequence at high speed and performing error correction. [Prior Art] In order to correct errors in multiple words using a BCH code or a Reed-Solomon code, it is necessary to generate a syndrome from received data and find the coefficients of an error locator polynomial. That is, this error locator polynomial is a polynomial having a value corresponding to the error position as a root, and by finding the coefficient of each term of the error locator polynomial, the position of the data in which the error has occurred can be calculated. Conventionally, Peterson, Berlekamp-Matsey, and Euclidean mutual division methods have been known as methods for determining the coefficients of such error locator polynomials. It is difficult to obtain suitable hardware, and when processing is performed by software, it is difficult to judge and there are problems with processing speed. In order to solve such problems, the present applicant previously proposed an error correction method and device for long distance codes that can decode long distance codes at high speed and is suitable for hardening. Patent Application No. 10016, filed on April 24, 1988, entitled "Error Correction Device"
No. 9). In the invention of this earlier application, when decoding a long distance code, each element Q1 . j to data word A (1°J-2) C, where 1≦i≦p11≦J≦p
+1, A0 to A2p-1 are syndromes or error positions], and the coefficients of the polynomial are determined by left-fundamental transformation of this matrix to form a unit matrix. [Purpose] The present invention aims to provide an error correction method for long distance codes that can decode long distance codes at higher speed than the error correction method described in the earlier application and is suitable for implementation in hardware. purpose. [Configuration] An example of the configuration of the present invention applied to a Reed-Solomon code of 256-element Galois field GF (28”), which is capable of 4-word correction and has a block of 255 words each consisting of 8 bits. Decoding of a Reed-Solomon code is performed through the following five steps: (1) Find a syndrome from the received data. (2) Find an error locator polynomial from this syndrome. (3) Find this error locator polynomial. (4) Find the error pattern from the error position. (5) Perform error correction using the error position and error pattern. In the present invention, the error location polynomial is calculated from the syndrome in (2) above. , and (4) calculation of the error pattern from the error position at high speed using the same processing method. All divisions (denoted by /) are modulo 2.
This calculation is executed by referring to the table in step 119. Assuming that the value of each word of the received data containing an error is rO-r254, the received data R(X) is given by the following equation (1). R(X) = r(, x254+ rl x2S3+ten r253 X' + r 254
``''''''''(1) The syndromes 81 to S7 generated from this received data are α0.α1.
These are the following equations in which α7 is substituted for each. So ”R(α') = ro + r 1 +--
−+ r2s+ 10S, =R(α') −r. ex
"+r, a2S" 10+r 2ss U'
+ r 2S4S2-R(α2)-r. α254°2
+r1α253°20 +r253 α1°2
10r254S7 =R(α7) −ro a”7
+r, a253.7+-""'-r 253 α
1°'+r2s<error locator polynomial σ(×) is as follows (3)
It is defined as Eq. (7(X) = (X+V1) (X0V2)
(x+V3)(x+V4) =X' +(73x3 +62 X2+σ, X+60
Here, V1 to V4 indicate the error position, that is, the word position with an error, and correspond to the value of the power of X in equation (1).
The word position in 0-decimal representation is required,
By referring to this table, when V=1, the lowest r25. term, α2 when V=4
= 4, it becomes like the term r 252. In this example, 8 types of syndromes of So''-37 are used, and the 8 unknowns representing 4 error positions and 4 error patterns are calculated by using these 8 types of syndromes. Although it can be calculated,
If the error pattern is determined first, the error position may not be calculated if there is the same value, that is, the same error pattern, so it is common to determine the error position first. Also, since this error position indicates the position of 255 words, it is a value from 0 to 255 (8 binary digits), and the error pattern is the error pattern within 8 bits of one word.
25 from “00000001” to “11111111”
There are five patterns, and errors are corrected by adding this pattern to the word in which the presence of an error has been detected. A relationship such as the following equation (4) holds between the values of the coefficients σ0 to σ3 of equation (3), which is the error locator polynomial, and the syndrome 5o=37. If the right side of equation (4) is shifted to the left side and the form of the matrix is changed, the following equation (5) is obtained. When both sides of this equation (5) are multiplied by a certain matrix A from the left and transformed into the following equation (6), σ0 to σ3 are obtained as shown in (7). O σ3”b415 σ2=b314Xσ3+b3,5 σ1°b2+3xσ2 +k)2.4×σ3+b2.s
σO=b++2XσI +bI+ 3×σ2+b,,,
Xσ3 +b +, s (7) When such a transformation is performed, the calculation is completed with 36 multiplications and divisions in the present invention, whereas 40 multiplications and divisions were required in the earlier application mentioned above. Therefore, the processing efficiency can be improved by 10%, and the processing speed can therefore be reduced by a corresponding proportion. The value of this left basic deformation is 0°15 respectively.
．． The cases of 85, 115, 193, 115, 161 and 231 will be specifically explained as examples. The first term on the left side of equation (5) is expressed by equation (8) below. The first step of transforming equation (8) into equation (6) is to transform the equation so that its first column becomes the first column of the first term on the left side of equation (6). Since the value in the first column and first row of equation (8) is 0, in order to replace the row in which the first column is not 0, swapping the second row and the first row results in the following equation (9). . Next, in order to set the value 15 in the first column and first row to 1, a modulo 2 calculation is performed to divide the values in each column of the first row by this 15, resulting in the following equation (10). Next, in order to set the value in the first column of the third and fourth rows to 0, for the third row, multiply the value in the first row by 85, and then add modulo 2 to each position in this third row. For the fourth row, multiply the value in the first row by 115, and then add modulo 2 to each position in the third row to obtain the following (1
1) Obtain the formula. Next, as a second step of transforming the above equation (8) into the form of equation (6), processing is performed to make the second column. Here, X is an arbitrary number. Since the value in the second column and second row is 15, the following equation (12) is obtained by dividing the second column by the value 15 in the second row, modulo 2. Next, in order to set the values in the third and fourth rows of this second column to 0, multiply the values in the second row by 87, add the third row and modulo 2, and By multiplying the value by 58 and adding the fourth line and modulo 2, the following
) to obtain the formula. In order to transform it into
By performing the division, the following equation (15) is obtained. Next, in order to obtain the fourth column, the fourth row is divided by modulo 2 by 101 of the fourth column, and the following equation (16) is obtained. With this, σ3~σ. As explained in equation (7), the value of σ3=b4. s = 15 σ2”b314Xσ3 +b3.s = O+54=5
4σ+ =b2+3 ×σ2+b214Xσ3+b2,
5=15x54+ 107x15+36=120σ0:
bl, 2×σl +bl+ 3×σ2+, 4Xσ
3+bl, 5 = 15x 120+ 107x54+36x15+10
It can be obtained sequentially as 7=64. If you solve this equation (17) using Cheno's algorithm and find its root, you can detect the error position.
In the case of this equation (17), x=1.2.4.8(18
), so as mentioned above, r2s4 ! in equation (1).
r2s3! There is an error in the term r2s□ and the term r251. Now that the error position has been detected as described above, a method for detecting an error pattern, that is, a correct data pattern will be described next. The error pattern at error position V1 is Yl, the error pattern at error position V2 is Y2, and so on, error positions V1, V2, V3
, V4 are error patterns Y1, ¥2, Y3, Y4, respectively.
shall correspond to At this time, error patterns Y1 to ¥4, error positions V1 to V
4 and syndromes S and -s3, the following relationship holds true. If the right-hand side of this equation (19) is transferred to the left-hand side and the matrix form is changed in the same way as the conversion from equation (4) to equation (5), the following equation (20) is obtained. In the left matrix of the left side of this equation (20), the above error positions are V1=1, V2-2, V3=4, V4-8.5o=0
．． The error pattern is obtained by substituting 51=15.52=85.53=115 and performing the same transformation as explained in equations (8) to 15), and the result is the following ( 21) It becomes as shown in the formula. Yl = 1 ¥3 = 1 Y, -1 (21) This detects that all error patterns are 1, so correct pattern 1 for the 1st, 2.4.8th word with an error. All you have to do is Note that the value of this error pattern Y represents the error pattern in decimal notation, and the above pattern 1 is "0000".
0001'' to the word at the error location modulo 2.
This is shown in the flowchart in Figure. In this flowchart, 5o-82t-1 has the syndrome and a12. represents each element of the matrix expressed by the following equation (30). (30) Figure (a) shows the process of assigning the value of syndrome 5o-37 of the first term on the left side of equation (5) to each element al, J. 1], the upper left element a1 of this matrix
, 1, and in step [2] add (i+j-
2) as a subscript, i.e. (i+j
-2) = (1+1-2) = Substitute the syndrome So with O as the subscript, specify the element a1.2 of the next column in the same row in step [3], and set the element in the row in step [4]. After confirming that the number does not exceed 5 in this case, return to step [2] and assign syndrome S1 to this element al+2, and in the following step [4], the assignment to the elements in this row is completed. Similarly, steps 82 to S4 are performed manually until . When the assignment of this row is completed, the result of the judgment in step [4] will be y, so in step [5]
Move to line 1 and substitute elements a2,1 to a2,5 in the same way as above, and a22. After completing the assignment, move to the third row and repeat the above (3) for all elements in the 5th row and 4th column.
0) Substitute m of the syndrome in equation. When all of this assignment is complete, the value of formula (8) has been assigned to the value of each element.■
The matrix in is shown in equation (32) below. l151931151612311 (32) Note that the subscript of S (i + j
-2) corresponds to the fact that the same syndrome is assigned to the elements on the diagonal line, as is clear from equation (31). In the same figure (b), in order to set the first column to [1000E, first, the first and second rows of the above equation (32) are swapped, and the following equation (33) corresponding to the above equation (9) is obtained. This figure shows a flowchart for performing the transformation. In step [7] of Fig. 1, etc., the column number P
Set to 1, and in step [8] set the row number to 1, which is equal to the column number.
and specify the elements al and l,

Check the value of this element in step [9], and if the value is 0, add 1 to the row number in step [10], and identify the row number in step [11] to calculate the number of rows up to the bottom row of the matrix. Search for rows within the range where the value of the element in the first column is not 0. If a row is found where the value of the element in the first column is not O, the process moves to step [12], and if the first row number is other than the initial value P=1,
In other words, if the value in the first row and first column is O, step [1
3], specify the element in the first column of the row to be moved to the first row by setting the subscript indicating the column number of the element to 1, and then temporarily set one word in step [14] to swap the rows. al in the save register d, which has a storage capacity of one card;
Move the value of element 1 and then perform the above steps

[9] ~ [
The element aLP,j in the first column of the row to be replaced with the row number LP detected in [11] is substituted, and the element a21.j is saved in the save register d. The value of a51.
Move to j. By repeating this process in step [15] while incrementing the number of columns/3 to be processed up to the last column, the state of ■ having the arrangement of formula (33) is reached, but the element a18 in the first column of the first row ．． If the 1st shift of is originally not 0 as in equation (33), there is no need to replace it, so we can move directly from step [12] to the state of . Steps [17] to [20] in FIG. 1(C) are performed as described above (
33) In order to set the value of the element in the first row and first column of the formula to 1, each element from the second column onwards in this row is divided by the value of the element in the first row and first column. , (34) is a flow that is modified as shown in the above-mentioned equation (10). Step [17] is to specify the second column element,
Add 1 to the subscript indicating the column to the row number P, and step [18]
Then, divide by the value of the element in the first row and first column of these elements,
By repeating steps [19] to [20] until the processing for each element in this row is completed,
The state of equation (34) above is obtained. In the next step [21], steps [17] to [20]
is completed and the following (35), which corresponds to the above equation (16), is obtained.
When the formula is obtained, the steps shown in the flowchart of FIG. 1(C) are steps for escaping from ■. In the next step [22], from the above equation (10), (1
In order to transform equation 1) or equation (12) into equation (13), 1 is added to the row number and column number. The next step [23] is the row P that should be set to the required value.
The value of the element in the same column is multiplied by the value of the Pth column of the row to which it belongs, and the value of the element itself is added, and this is repeated until the end of the row by steps [24] and [25]. After processing, step [26], [27]
The next line is processed by In the next step [28], the next column is set to a predetermined value,
For example, in order to perform the process of setting the second column to [0100E, 1 is added to the column number and the process returns to step [8] while unprocessed rows remain. In addition, ■ is a case where there are few errors, etc., and the above step [1]
1] shows the route when the process is already unnecessary. Step 30 where all processing is completed in this way
The matrix obtained in the previous stage of ] was previously shown as equation (35), and is shown here again. The next steps [30] to [40] are for performing the arithmetic processing shown in equation (7) above to obtain the coefficient σ, and this equation (7) is recited here as equation 36). σ3”b415 σ2−b11.4Xσ3+b3,3 σ 1 : b 2. ri × σ2 + b
2.4 X σ3 +b 2. 5σ. =b,
, 2xσ, tensu, , 3xσ2+b, , 4xσ3+b+,
s (36) In step [30], the coefficient σ to be calculated by calculation, that is, the subscript of σ on the left side is set. In the next step [31], d is substituted into the rightmost term on the right side of each equation (36) above, and in step [32]
Then, K=3. When σ3 with P=5 is obtained and the process is completed, a decision is made to jump to step [37], and depending on the result, the process moves to the next step [33]. Step [33] sets the subscript of σ on the right side of equation (36) above, and step [34] multiplies σ specified by and the value of register aK+l+L and adds the value to d above. It is assigned to d again. In step [35], 1 is added to L and the result is checked in step [36]. If L is equal to (P-2), the process is terminated, and if not, in step [34]
] and [35) are repeated. In step [37], σ has already been calculated as a variable of d, so d is set to σ. Assign to . In step [38], when K is decreased by 1 and the subscript of σ is positive or 0, the calculation of σ has not been completed, so the processes of steps [31] to [37] are repeated. In step [40], 1 is substituted for σP-1 according to the definition of the error locator polynomial, and the process is completed. FIGS. 2 to 4 each show the configuration of an apparatus for implementing the error correction method according to the present invention. FIG. 2 uses a configuration similar to that of a conventional error correction device, in which a bus 30 includes a CPU 31, a RAM 32, an RO
M33 and ■10 port 34 are respectively connected, and the CPU 31 reads programs and tables stored in the ROM 33 and stores them in the RAM 32.
Processing including modulo 2 calculation is performed based on this program and table, and the syndrome obtained from the received data word is sent to the first I10 port 34.
The error position and error pattern are inputted manually from the CPU 31 and output from the ■/○ ports 34, respectively, and are used to correct the decoded data word. In the configuration shown in FIG. 3, the calculation of modulo 2 is performed using CP
Since the processing is performed in U, there is a drawback that the calculation time is long and the processing speed is low. FIG. 3 shows an embodiment of the present invention that solves the above-mentioned drawbacks, in which a CPU 41, a RAM 42, a ROM 43, and an I10 port 44 are connected to the bus 40, and furthermore, according to the present invention, a modulo 2 multiplier 451, a modulo 2 multiplier 451, A modulo 2 arithmetic processing unit 45 shown as including a divider 452 is provided, and the modulo 2 arithmetic processing unit 45, which is shown as including a divider 452, is provided, and performs steps [14], [18], and [23] in the flow chart of FIG. By having the calculation processing section 45 perform calculations, the processing speed can be improved. FIG. 4 shows an apparatus in which the arithmetic processing of the flowchart shown in FIG. 1 is mainly performed by transfer.
A division unit 54 for performing the division in step [18], an arithmetic unit 55 for multiplication and addition for performing the processing in step [23], and a saving necessary for exchanging data words in step [14]. A register 56 is connected to the bus 50, and this bus includes a RAM 52,
○A transfer control unit 51 including a CPU and mainly for controlling data transfer is connected together with the port 53. The division unit 54 and calculation unit 55 in this embodiment correspond to the calculation processing unit 45 in FIG.
6 is the latch 5611, 5612 and the 3-state buffer 5
The data words are exchanged by storing two data words to be exchanged in the respective latches 5611 and 5612 and sequentially reading them out. In the embodiment of the error correction method of the present invention described above, the matrix is transformed into an upper triangular matrix so that all the diagonal elements are 1, but it is not necessary that all the diagonal elements are 1. 2nd row, 1st column, 3rd row, 1st row. As long as the elements below the diagonal elements, such as the second column and the fourth row, first column to third column, are all 0, they may be any numbers other than O, which are different from each other. A second embodiment that performs such processing will be described below. The equations corresponding to the above equations (4) and (5) can be expressed as the following (40
). It is shown in equation (41). σ3 = 1/Ca, 4 X (C4, s) σ2 = 1
/C3,3X (C3,4Xσs +C3,s)σ+
=1/C2,2×(C2,3×σ2 +C214Xσ
3+C2,S) σ. -1/C+, + X (C+, 2 Xσl
+c,, 3 x σ2 + C, 4 ~S7 values are respectively 0, 15, 85, 115°193
The cases of 115.161 and 231 were taken as examples (42
) will be specifically explained using the formula. As a first step in transforming this equation (42), its first column is transformed so that it becomes. Here, xl is any value other than 0. Since the value in the first column and first row of equation (42) is O, in order to replace the row in which the first column is not 0, the second row and the first row are swapped, resulting in the following equation (43). . Next, in order to set the values in the second row and below of the first column to 0, multiply the values in each column in the first row by 85/15 and add them to the values in the corresponding columns in the third row, and , 11571 for each column value of the first row
Multiplying by 5 and adding it to each corresponding column value in the 4th row gives us
The above matrix is transformed as shown in equation (44) below. Next, processing is performed such that the second column of the above equation (44) is obtained. Here, x2 is any value other than 0, similar to xl. Since the value in the second column and second row is 15, which is not O, the number (43)
Similar to the transformation from formula to formula (44), the values in each column of the second row are multiplied by 87/15 and added to the values in the corresponding columns of the third row, and each value in the first row is By multiplying the column value by 58/15 and adding it to the corresponding column value of the fourth row, the matrix of the above equation (44) is transformed as shown in the following equation (45). Next, in order to transform the third column of the above formula into By multiplying the values by 0/101 and adding them to the values in the corresponding columns of the fourth row, the matrix of equation (45) above becomes the following (4
6) It is transformed as shown in Eq. Note that x3 is any value other than 0, similar to X, , X2. Next, by performing the calculation shown in equation (41),
σ3～σ. is calculated as follows. σ3 = 1/C4,4X (C4,s) = 1/10
1 X41=1+σ2 ”1/C3,3X (C3,4
Xσ3 +C3,S)=1/l0IX(0,X15+
97) = 54σl = 1/C2, 2X (C2,
3Xσ2 +C214Xσ3+C2,5) -1/15x (58x54+ 115x15+193)
=120 σ. -1/C+, + x (C+, 2 Xσ, +C
,,3Xσ2+C,,4Xσ3 +C+,5) =1/15X(58X 120+115X54+ 19
3X15+115) =64 Naturally, σ0~ obtained in this way
The value of σ3 is σ determined in the first example. It matches the value of ~σ3. In addition, in this second embodiment, the calculation for setting the diagonal elements 1, C2, 2, C3, 3, and C41 of the equation (40) to 1 is unnecessary, but the above σ Since it is necessary to divide by the values of these diagonal elements when calculating , this is the same as the first embodiment in terms of the number of calculations. FIGS. 5(a) to 5(d) are flowcharts showing the processing of this second embodiment, and FIGS.
Since the processing is the same as that shown in (a) and (b) of the figure, the explanation will be omitted. In the state indicated by ■ at the beginning of the flowchart in FIG. 5(C), the matrix shown in equation (42) above is generated on the memory of the processing device, and this is flexible as equation (50) as shown below. . The flowchart in FIG. 5(C) corresponds to the flowchart in FIG. 1(C), but step [17] in FIG.
] to [20], there is no step corresponding to step [1
01:I, [102) and [: 104) ~ [
: 108] are the steps [: 2
1:], [22:] and [23] to [27]. In step C103:], from the above equation (43), (44)
Multipliers 85/15 and 115/15 when transforming into Eq.
is calculated, and in the next step C104], Fig. 1 (
In step [23] of C), al+ J ”” al+ J +aP+J Xar-
The calculation of p was performed, but instead of this al, , d obtained in step (103) is used to calculate ai,, = attj ten ap,, x (1). It is different from step [23] in C). Fig. 5(d) is a flowchart corresponding to Fig. 1(d), and steps [109) to [: 12] in Fig. 5 are different from step [23] in Fig. 5(d).
1] corresponds to steps [28] to [40] in Fig. 1(d), but instead of the assignment σ, = d in step [118], the assignment σ id / ax, x is made. They differ in what they do. Although the above explanation is about the Reed-Solomon code on GF (28>, the present invention is not limited by the Galois field or the size of the Galois field, and also uses b-adjacent codes etc. as correction codes. It is obvious that the present invention can be applied to cases where codes other than Reed-Solomon codes are used. [Effects] According to the present invention, the solution to the determinant is not obtained by direct calculation, but by deformation of the determinant. The present invention is the same as the earlier application mentioned above in that a solution is obtained by calculating the data words constituting the matrix, but since the number of calculations is further reduced than in this earlier application, there is no need to place an excessive burden on the arithmetic unit. The special effect of further speeding up the processing can be achieved.

[Brief explanation of the drawing]

FIG. 1 is a flowchart of an embodiment for calculating the coefficients of an error locator polynomial using the error correction method of the present invention; FIGS. 2 to 4 are embodiments of an error correction device to which the error correction method of the present invention is applied; FIG. 5 is a flowchart of another embodiment of calculating the coefficients of the error locator polynomial using the error correction method of the present invention. Patent Applicant Ricoh Co., Ltd.] Flow jet Figure 1 (b) (C) Blow loaf 4-) Figure 1 (d) Blow valley Figure 1 Broach feed Figure 5 (b) 5 Figure (d) Broochie Y Figure 5

Claims (4)

[Claims] (1) When decoding a long distance code, in order to obtain the coefficients or error patterns of each term of the error locator polynomial from the syndrome, data words A_(_i_+_j_-_2_ ) [However, 1≦
i≦p, 1≦j≦p+1, A_0 to A_2_p_-_1
An error correction method in a long distance code, characterized in that the coefficients of the polynomial are determined by setting a syndrome or an error position] and performing left fundamental transformation on this matrix to transform it into an upper triangular matrix. (2) When decoding a long distance code, in the case of n errors that do not exceed the maximum number that can be corrected by the long distance code, use the above p to calculate the coefficient of each term of the error locator polynomial from the syndrome. Set the syndrome S_(_i_+_j_-_2_) as the data word for each element a_i_, _j of a matrix of n rows and (n+1) columns, where the number of errors is n, and transform this matrix into an upper triangular matrix by performing left fundamental transformation. 2. The error correction method in a long distance code according to claim 1, wherein the coefficients of the error locator polynomial are determined by calculating the error position polynomial. (3) When decoding a long distance code, in order to find an error pattern corresponding to each error position using the error position and syndrome, element a_ of the n rows and (n+1) matrix where p is n
V_i＾j＾−＾1 [
However, 1≦i≦n, 1≦j≦n, V_1 to Vn are error positions], and the syndrome S_i_−_1 is set as the data word of elements a_i_, _n_+_1, respectively, and this matrix is subjected to left fundamental transformation. 2. The error correction method in a long distance code according to claim 1, wherein each error pattern is obtained by transforming the matrix into an upper triangular matrix. (4) When the number of elements that require calculation is small, by adding rows and columns in which the data words of all elements are 0, the same processing can be performed without changing the arrangement of elements. An error correction system in a long distance code according to claim 1, characterized in that: