JPH01136424A - Decoding system for reed solomon code applying error correction/error detection/missing error correction simultaneously - Google Patents

Abstract

PURPOSE:To apply error correction/error detection/missing error correction simultaneously by using a decision equation comprising relating equation of a correction syndrome if any missing error exists and regarding an error of a number to be error correction and allowing it to be remained to be error detection if number of errors more than that. CONSTITUTION:Coefficients sigmad1, sigmad2 are obtained by a logarithmic function calculation section 1 and T0-T2 are obtained by a correction syndrome calculation section 2 from them and syndromes S0-S4. A decoder 3 for a single error correction reed Solomon code corrects a single error by the T0-T2 and sigmad1, sigmad2 to obtain the error location x1 and the quantity of error e1'. Then the section 6 of the syndrome obtains S0, S1 and a missing error quantity calculation section 7 calculates d1, d2 by using them. When the output of an inverter circuit 5 goes to '1', it is decided to be the presence of a duplicated error and only the error detection is implemented.

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention is directed to a Reed-Solomon code decoding system that simultaneously performs "error correction/error detection/erasure yield correction".

There are two possible types of errors: normal errors, in which both the location and size of the error are unknown, and erasure errors, in which the location and size of the error are unknown, but the size of the error is unknown.

Currently, duplicated Reed-Solomozi codes are often corrected by combining erasure errors whose error locations are known and normal errors. In such a case, if corrections are made to the Kerr Cup error correction capability, erroneous corrections will occur and the reliability will decrease, so various measures are taken to prevent erroneous corrections. If the error detection is stopped when the constant term of the error locator polynomial is non-zero, 3.4 errors can be detected.

However, since this method cannot detect 3.4 errors 100%, 3.4 errors that have slipped into the decoding process for errors of 2!1 or less are removed as undecodable. This is not a smart method, and it is also impossible to detect only 3 errors. Therefore, we present a new decoding method that overcomes these drawbacks and simultaneously performs efficient error correction, error detection, and erasure errors.

Now, the lead φ soton code is defined by the following generator polynomial.

However, d is the minimum distance of the code First, we will outline the conventional Forney decoding method.

The decryption steps are as follows.

Step l: Calculation of syndrome.

Step 2: Find the symmetry number of the location of the erasure error.

Step 3: Find the corrected syndrome using the syndrome and the symmetric number of erasure error positions.

Step 4: Calculate the coefficients of the error locator polynomial and determine the number of errors accordingly.

Step 5: Calculate the w4 position.

Step 6: While correcting the syndrome, consider all errors as erasure errors and calculate the magnitude of the error.

Next, we will discuss only the points that change with the new decoding method.

Step 4': It is determined whether the number of errors is less than or equal to a certain value or more than a certain value using a determination formula consisting of an equation between the correction syndromes.

Step 5': Find the "positions and sizes" of "normal errors" all at once.

Step 6': Calculate all magnitudes of two erasure errors using the same formula, which is simpler than Forney's method.

Therefore, it is possible to parallelize the hardware circuit configuration.

Now, steps 4' and 5' are based on the following principle. That is, the number of normal errors is t, and the size of the error is e4.
, the position of the error is XI, and the polynomial whose root is the position of the erasure error is σd(Z), then the modified syndrome is expressed as follows.

However, E+=ej(ij)”・cx<xノ> (3
) Therefore, the error size E1 and the error position x can be regarded as normal errors and decoded. Therefore, the error determination formula can be obtained by treating the modified syndrome 7n like a normal syndrome. Furthermore, when there are relatively few erasure errors, it is efficient to find E+ using a normal decoding method using the modified syndrome, and then find the actual error magnitude e+ using equation (3).

Two specific examples are shown below *melto.

(1) "Single and double error correction 73 double error detection/single erasure error correction" Reet "Solomon code decoding algorithm In this case, d = 7, and 3!!L error correction Reed So a
It is the same as the t code. aJ simply indicates the decoding method.

Step 1: Calculation of syndrome S, (0≦intestinal≦5) Step 2: If the erasure error position is yI, σd(z)
=z+σaI, c d+ =y I(4) Step 3
:1! Positive syndrome Tn"Sn-++y+Sn, O≦n≦4 (5)
Step 4: The determination formula for "single and double error correction/multiple error detection" when erasure errors are excluded is as follows.

E 1M3=T @T2T a+ T2”+ T I2
74+ When Tl1732E1M3 is zero, it can be determined that there is an error of double or less, and when it is non-zero -, it can be determined that it is an error of 311 or more. When there are no more than three errors, this discrimination ability is 100%.

Step 5: Correct 2 plate errors or less.

A=T@T2+Ti2 (7) When equation (7) is zero, it is a single error, and when it is non-zero, it is a double error.

In the case of a single error, the error position x + = T + / Ta (8) E + = T s The magnitude of the error σa (x) In the case of a double error, from equation (10) x 2 + cr 21 X + a 22 =
o +7) Calculate the root (Bolkienhon's method is efficient) to find the location of the error. Next, E
+.

Find E2 using the following formula.

Therefore, Step 6: Correcting the syndrome ・Ss'=S
s+e+'x+'÷e2x2 (+3) Figure 1 (1 ) The part of the decoding algorithm for "single/double error correction" in Figure 1. (1) is shown in Figure 1 (2).

(2) "Single error correction/double error detection 73 double erasure error correction" Reed-Somoji code decoding algorithm This code also has d=7 and is decoded in the same way as in the previous section, and the decoding algorithm is shown in Figure 3. shown. yI (I=1,2°3) is the position of the erasure error.

The determination formula for "single error correction/double error detection" is as follows.

81M2=T・T 2+T 12
(15) That is, when the equation (15) is zero, it is determined that it is a wheel error, and when it is non-zero, it is determined that it is a double error. The magnitude of the erasure error is calculated using the following equation, which is simpler than Fourny's and is also the same.

(i = 1.2.3) (3) "Single error correction/2f [error detection/2!l erasure error correction Reed-T-T code decoding algorithm] The minimum distance of this code is d = 6, Single/double error correction/3
It is the same as a heavy error detection code.

By slightly modifying the decoding algorithm in section (2), a decoding algorithm for this code can be obtained, as shown in FIG.

The magnitude of the erasure error is calculated using the following equation, which is simpler than Fourny and the same as in section (2).

Similarly, various decoding algorithms for "error correction/error detection/erasure error correction" are derived.For example, the determination formula for "single/double error correction/3Φ4 error detection" in step 4 is as follows. .

E 2M4= [(T0T 2Tn+T 2”+Ti
'T 4◆T s T 52=o)ND (Ti2(T2T4+T 3')+T a"(T @T
4+T22)+T42(T@T2+Ti”)=O)
] (17) When E2M4 is true, double or less error,
When it is false, it is determined as a 3/4 error. In addition, the judgment formula for "single error correction 72φ3 error detection" is E2M3= ((T0T2+Ti2=O)AND (T @T 32+T 23=O)) (
18) When E2M3 is true, it is determined that the wheel is wrong, and when it is false, it is determined that 2/3 is wrong.

The above algorithms are easily implemented in software (including microprogramming). Currently, a commonly used method is to perform microprogramming using an LSI dedicated Galois rest field processor. Galois field operations are performed using exponential representation or vector representation, and the implementation method is already known.

Next, a diagram showing the principle of a decoder in which the above algorithm is implemented in hardware (including a full custom LSI) is shown. In this case as well, Galois field operations are performed using exponential representation or vector representation, and the implementation method is already known. Therefore, the principle diagram of the decoder shown below can be easily realized in hardware.

Figure 4 shows "single/double error correction/73 double error detection/single erasure error correction", "single/double error correction/triple error detection", and "single/double error correction/3j1. FIG. 2 is a principle diagram of a multifunctional Reed-Solomon code decoder with "quadruple error detection"; Next, three functions will be explained.

(b) The decoding operation of "single/double error correction/311r4 error detection/single erasure error correction" will be explained. first,
- The correction syndrome unit 1 calculates Ts to T4 from the syndrome 5e=Ss and the erasure error position y1. In this case, the syndrome switching unit 2 of SneA outputs Ts=74.

Using this, single and double errors are corrected in the "single/double error correction" Reed-Solomon code decoder 3 (this decoder is already known), and error positions x1, x2 and El
, E2 is output. In addition, T@=Ti =T2=T3=
When Ta × Goose = x2 = E + = E2: zero element, A = T
@When T2+Ti2=O, there is a single error, so x2=E2=
Let it be zero yuan.

Next, the error magnitude calculation unit 4 calculates error magnitudes e+' and e2' from El and E2. 5WC12 is on the lower side. Then, the erasure error magnitude calculation unit 6 calculates the erasure error magnitude dl.

Furthermore, using Ts-74, “single/double error correction 73
The "heavy error detection" determination unit 6 outputs 1 when ElM3=0 in equation (6) is established, which is inverted by the inverter circuit 8 and outputs O.
Send O to the R circuit 10. Therefore, the error detection signal is also O
It is. If ElM3≠0, the error detection □ output signal becomes 1, and the above decoded output is invalidated and error detection is performed.

(b) When performing normal "single/double error correction/triple error detection" decoding without considering erasure, the output of the SW A syndrome switching section is set to 5sxSa.

Using this, single and 2m errors are corrected in the "single/double error correction" Reed-Solomon code decoder 3, and error positions XI, X2 and El, E2 are output. 5WC12 is on the upper side, and El and E2 are output as they are.

Furthermore, using 5a=Sa, “single/double error correction/3
The "multiple error detection" determination unit 6 performs error determination in the same manner as in section (a).

(c) The operation when decoding "single/double error correction/3!/quadruple error detection" is performed is the same as in item (b).

However, in the additional part for determining "single/double error correction/triple/quadruple error detection", the lower half of formula 17 E'2M4'=Tt2(T0T4+T32)+T42(
T0T4+T22)+T42(T0Tz+Ti2)=
A determination of O(19)□ is made, and if the equation holds, l is output.

5WBII is on the upper side, and when the output of the OR circuit 10 is 1, E2M4 in equation (17) is false, so 3
- Limit the error detection to a quadruple error.

FIGS. 5(+) and 5(2) are diagrams showing the principle of a Reed-Solomon code decoder with "single error correction, double error detection, and triple erasure error correction." First, in the symmetric space calculation unit 1, σdl, σd
2. Find σd3 and use this and syndrome 5Q4S! The corrected syndrome calculation unit 2 calculates T@, T electric, Ta from * Ta, Ti, Ta and σd1% σd2, σ63
Accordingly, the decoder 3 of the single error correcting Reed-Solomozi code corrects a single error and determines the error position x1 and error magnitude elo. However, when Ta:Ti=72=0, xI=e+'=zero element. Then, in the syndrome correction section 6, 50Z5. ', 5.1 are obtained, and using these, the erasure error magnitude calculation unit 7 calculates d1, da, and da. Note that the "single error correction/double error detection" determination unit 4 outputs l when εu12=o in equation (15), and outputs 0 when the equation does not hold.

When the output of the inverter circuit 5 becomes 1, it is determined that a double error has occurred, and the error detection is stopped.

FIG. 6 is a principle diagram of a "single error correction 72 double error detection 72 double erasure error correction JI7-)" Solomon code decoder.

The operation is similar to that shown in FIG.

First, the symmetry interval calculation unit 1 calculates σ-1 and σd2, and the modified syndrome calculation unit 2 calculates 1 unit, TI, and T2 from these and the syndromes Sa to S4. With T, TI, T2 and σdlsσd2, the decoder 3 of the single error correcting Reed-Solomon code corrects a single error and determines the error location x
I, find the error magnitude el'. However, Ts=Ti
When =Ta=O, x+=e+'=zero element. Next, the syndrome correction unit 6 calculates Ss' and S+',
Using these, the erasure error magnitude calculation unit 7 calculates dl, d2.
Calculate. The "single error correction 7 double error detection" determining unit 4 outputs 1 when E1Ma=O in equation (15), and outputs O when the equation does not hold. When the output of the inverter circuit 5 becomes 1, it is determined that a double error has occurred, and the error detection is stopped.

Next, an application example of the decoding method and decoder described above will be shown.

Currently, DAT (digital audio tape) uses dual codes such as a double error correcting Reed-Solomon code and a triple error correcting De Solomon code. Therefore, when decoding a triple error correction Reed-Solomon code (d=7), in order to remove error corrections, (1) "single, 2m error correction 73 triple error detection/ "Single erasure error correction" Reed-Solomon code decoding algorithm (
2) Single error correction, double error detection, triple erasure error correction, Reed-Solomon code decoding algorithm and Figure 4:
Figure 5: Decoder for multi-functional read Φ SOMON code with "erasure error correction", "single/double error correction, 73 double error detection", and "single 4 error correction/311/4m error detection".
"Single error correction 72 double error detection/3m erasure error erasure correction"
A Reed-Solomon code decoder can be applied.

It is possible to decode ``single error correction 72 double error detection/single erasure error correction'' Reed-Solomon codes for 2ji error correction De Solomon codes, including CDs (compact discs). However, this decoding is performed according to FIG. 3 of the present invention.
Since it is only necessary to slightly modify FIG. 6, the explanation will be omitted.

Next, two examples of duplex codes and their decoding methods, which are particularly effective for decoding according to the present invention, will be presented.

(1) “Single/Double Error Correction 73 Double Error Detection” REIT
・Double code of Solomon code (d=6) and triple error correction ・Double code of de Solomon code (d=7) This code is shown in FIG. Multiple error detection: Decodes the de Solomon code, detects triple errors or more, and sets a flag. Next, C2, that is, the triple error correction Reed-Solomon code, is
)--Decodes as a Solomon code, corrects errors of double or less, detects triple and quadruple errors, and sets a flag.

Repeat this one or more times alternately with 01 and C2.

(From the second time onward, there is also a method of decoding only those with flags set.) The flags decrease.

If it is done by software (including microprogramming), currently it is done about once with DAT, but the fourth method of the present invention
It can be repeated many times using the decoder shown.

After the final decoding of C2, furthermore, for C2, “
Single/Double Error Correction 73 Multiple Error Detection/Single Erasure Error Correction Decodes the Reed-Solomon code, that is, treats one of the flags as an erasure error, and can correct up to 3 erasure errors when there are only erasure errors. It is.

In order to perform the above decoding at high speed, it is sufficient to switch the three functions of the decoder shown in FIG. 4 for decoding. Therefore, decoding is possible with one decoder.

Also, if there are many erasure errors, "Single error correction 72" in Figure 2
The Reed-Solomon code decoding algorithm of "multiple error detection, 73 double erasure error correction" or the "single error correction, 72 double error detection, 73 double erasure error correction" Reed-Solomon code decoding algorithm shown in Figure 5 is used. At this time, if there are only erasure errors, up to four erasure errors can be corrected.

Of course, all of the above decoding can be performed by software (including microprogramming), but the error correction ability improves as the number of repetitions of decoding C1 and C2 increases using a high-speed decoder.

(2) "Single 4 error correction/3m error detection" duplication code of Reed-Solomon codes This code is shown in Figure 8 -C
Both I and C2 are Reed-Solomon codes with single/double error correction and 73 double error detection, and d=6. In this decoding, first, a C1, ie, "single/double error correction 73 multiple error detection" Reed-Solomon code is decoded to correct errors of double or less, and to detect triple or more errors and set a flag. Next, C2#chi “single 4 error correction/triple error detection
The V-Solomon code is decoded to correct errors of 211 or less, and triple or more errors are detected and flagged. This decoding can be repeated one or more times. This can be done by software (including microprogramming) or by a high-speed decoder, and the method is already known (the principle is shown in Figures 1 and 41). !I).

Finally, the Reed-Solomon code decoding algorithm with "single error correction/double error detection/double erasure error correction" shown in Fig. 3 or the "single error correction 72 double error detection/double erasure error correction" shown in Fig. Decoding is performed using a corrected JI+4-Solomon code decoder.

[Brief explanation of the drawing]

Figure 1 (1) and (2) show "single/double error correction/triple error detection/single erasure error correction" Reed-Solomon code decoding algorithm. Detection/3!l Erasure Error Correction” Reed-Solomon Code Decoding Algorithm 3
Figure 4 shows "single error correction, 7 double error detection, 72 double erasure error correction" and "Leat"-Solomon code decoding algorithm. Principle of multifunctional decoder for de Solomon code with single-fist double error correction/multiple error detection and single-fist double error correction/multiple error detection. Figure 1: Modified syndrome calculation unit 2 : Syndrome switching section 3: Single/double error correction Reet'/Solomon code decoder 4: Error magnitude calculation section 5: Erasure error magnitude calculation section 6: Single/double error correction 73 Judgment 7: "Single/double error correction/31i-quadruple error detection" judgment 8.9: Inverter circuit 10: OR circuits 11, 12: Switches, Fig. 5 ( 1) and (2) are "single error correction 7 double error detection/triple erasure error correction : Single error correction Reed-Solomon code decoder 4:
"Single error correction/double error detection" determination section 5: Inverter circuit 6: Syndrome correction section 7: Elimination error magnitude calculation section FIG. JI7-)--Principle diagram of Solomon code decoder 1
:Symmetry number calculation unit 2: Modified syndrome calculation unit 3: Single error correction leit”◆Solomon code decoder 4
: "Single error correction/double error detection" determination unit 5: Inverter circuit 6: Syndrome correction unit 7: Elimination error magnitude calculation unit FIG. 7 shows "single/double error correction 73 double error detection" Figure 8 shows a duplication code of a Reed-Soton code and a triple error correction Reed-Solution code.
Duplicated code between Solomon codes Figure 1 (1) Figure 1 (2) Figure 2 Figure 3

Claims (1)

[Claims] 1. When there is an erasure error, using a judgment formula consisting of the relational expression of the modified syndrome Ti, "error correction" is performed up to a certain number of errors, and when there are more errors than that, "error correction" is performed. Reed-Solomon code decoding method 2 that simultaneously performs "error correction/error detection/erasure error correction" characterized by "limiting to detection", decision formula E_1M_3=T_0T_2T_4+T_2
^3+T_1^2T_4+T_0T_3When ^2 is zero, it is "error correction", and when it is non-zero, it is "limited to error detection". "Single/double error correction/triple error detection/single erasure" Reed-Solomon code decoding method that simultaneously performs "error correction" (when decoding is done by software including microprogramming) 3.
Reed-Solomon code decoder 4 that simultaneously performs "double error correction/triple error detection/single erasure error correction"; In a Reed-Solomon code decoder that simultaneously performs "erasure error correction", "single/double error correction/3
The E_2M_4 is operated as a Reed-Solomon code decoder for double error detection, and the E_2M_4
'=T_3^2(T_2T_4+T_3^2)+T_4
^2(T_0T_4+T_2^2)+T_5^2(T_
By operating 0T_2+T_1^2) in parallel, E_2M_4=[E_1M_3=T_0T_2T_
4+T_2^3+T_1^2T_4+T_0T_3^2
=0ANDE_2M_4'=T_3^2(T_2T_4
+T_3^2)+T_4^2(T_0T_4+T_2^
2)+T_5^2(T_0T_2+T_1^2)=0]
(However, Ti can be read as Si.) If it is true, it is "error correction", and if it is false, it is "limited to error detection". "Single/double error correction/triple error detection/single erasure error correction" and "single/double error correction/3 "multiple error detection" and "single/double error correction/triple/
When the decision formula E_1M_2=T_0T_2+T_1^2 is zero, the multi-functional Reed-Solomon code decoder 5 for "quadruple error detection" is set to "error correction", and when it is non-zero, it is "limited to error detection", and further , the magnitude of the erasure error can be calculated by Forney's method or by the "single error correction/double error correction" method, which uses ▲mathematical formulas, chemical formulas, tables, etc.▼(i = 1, 2, 3). Reed-Solomon code decoding method that simultaneously performs "detection/triple erasure error correction" (when decoding is done by software including microprogramming) 6. The Reed-Solomon code decoder 7 performs "error correction/double error detection/triple erasure error correction" simultaneously, and when the decision formula E_1M_2=T_0T_2+T_1^2 is zero, it is considered "error correction", and when it is non-zero, it is " Furthermore, the size of the erasure error can be calculated using Forney's method or d_1=(S_0'Y_2+S_1')/(Y_1+Y
_2), d_2=(S_0'Y_1+S_1')/(Y
A Reed-Solomon code decoding method that simultaneously performs "single error correction/double error detection/double erasure error correction" characterized by using A Reed-Solomon code decoder 9 that simultaneously performs "single error correction/double error detection/double erasure error correction" by realizing item (7) using hardware (including LSI), which can be realized in the same manner as above. What, judgment formula E_1M_2
When =T_0T_2+T_1^2 is zero, "error correction"
Reed-Solomon code decoding method 10 that simultaneously performs "single error correction/double error detection/single erasure error correction" characterized by "limiting to error detection" when non-zero, double error In duplex codes such as corrected Reed-Solomon codes and triple error-corrected Reed-Solomon codes, triple error-corrected Reed-Solomon codes are classified into single/double error correction/triple error detection/single erasure error correction. ” code, “single error correction/double error detection/triple erasure error correction” code, or “single/double error correction/triple/quadruple error detection” code. Decoding method 11 for double code using triple error correction Reed-Solomon code. First, decoding C_1, that is, "single/double error correction/triple error detection" Reed-Solomon code, detects triple errors or more. Detect and flag errors in Next, C_2
That is, a triple error correction Reed-Solomon code is decoded as a "single/double error correction/triple/quadruple error detection" Reed-Solomon code, correcting errors of double or less and detecting triple/quadruple errors. do. Sets a flag when an error is detected. This is C_
1, C_2 Repeat one or more times alternately. (From the second time onward, there is also a method of decoding only those with flags.Also, it is also possible to decode only by repeating this.) Then, after the last decoding of C_2,
Decodes the Reed-Solomon code (double error correction/triple error detection/single erasure error correction). In other words, one of the flags is treated as a loss error. Furthermore, if there are many erasure errors, a "single error correction/double error detection/triple erasure error correction" Reed-Solomon code is decoded. Encoding and decoding method 12 for double code of "single/double error correction/triple error detection" Reed-Solomon code and triple error correction Reed-Solomon code, which features the above decoding method, first, C_1, ie, "single/double error correction/triple error detection" Reed-Solomon code is decoded to correct errors of double or less, detect errors of triple or more, and set a flag. Next, C_2, that is, "single/double error correction/triple error detection" Reed-Solomon code is decoded to correct errors of double or less, and detect triple or more errors and set a flag. This decoding is repeated one or more times. (There is also a method of decoding only those with flags from the second time onwards. Also, decoding may be performed by just repeating this process.) Finally, "Single error correction/double error detection/double erasure error correction" ” Decodes the Reed-Solomon code. Encoding and decoding method of double code between "single/double error correction/triple error detection" Reed-Solomon codes featuring the above decoding method