KR20000024730A - Method for calculating syndrome of extended reed-solomon codes and decoding method thereof - Google Patents

Abstract

PURPOSE: A method is to calculate a syndrome necessary for decoding £2¬m,2¬m 2t|(wherein m means message polynomial and t means error correction capability) extended Reed-Solomon codes and to decode the £2¬m,2¬m-2t| extended Reed-Solomon codes using a syndrome calculation method. CONSTITUTION: A method for calculating extended Reed-Solomon codes comprises the steps of: (S410) calculating syndrome S0-S2t-1 for code C; (S420) determining error locations for the code C by using S0-S2t-2; (S430) going to S470 if errors are occurred below t-1 symbol in code C; (S440) determining error locations by further using syndrome S2t-1 for the code C; (S450) going to S470 if error is occurred below t symbol in code C; (S460) announcing failure of decoding and ending; (S470) estimating error values equivalent to error locations and calculating P0(x) by estimating code word c(x) of code C; and (S490) correcting error of code Cprime and ending the decoding.

Description

Method and Decoding Method of Extended Reed-Solomon Codes

The present invention relates to a method for error correction, and more particularly, to a method for obtaining a misrepresentation of an extended RS code and a decoding method using the same.

When data is transmitted or stored in a communication system or data storage system, various types of noise, distortion, and interference cause loss of information, or error. In order to properly resolve these errors, the use of error correcting codes is indispensable.

1 is a block diagram showing the configuration of a general digital communication system. Data generated in the data source 10 is encoded by the source encoder 12 and then provided to the channel encoder 16. The channel encoder 16 adds and encodes an error correction code for an error during transmission. The encoded data is modulated in the modulator / transmitter 17, transmitted over the channel 18, and the signal received by the receiver / demodulator 20 on the receiving side is recovered.

The data demodulated by the receiver / demodulator 19 is corrected for errors occurring during transmission by the decoding process of the error correction code. The error corrected data is decoded by the source decoder 22 and restored to the original data by the data synchronizer 20.

The error correction code can be largely divided into block codes and tree codes according to a configuration method. RS code (Reed-Solomon Codes), a typical block code, is a kind of cyclic codes.It is easy to calculate encoding and synthesis, and the decoding method is relatively simple and easy to implement. It is applied.

The RS code shows better performance in burst error than in random error and is used as the outer code of the concatenated code.

If p and m are each a random number and a positive integer, q = p ^m Finite field with four elements GF (q) Let's think. RS code is GF (q) If the element of is a symbol and the code length is n n = q-1 Is a non-binary code. Generally [n, k] The sign indicates that the sign length is n and the dimension of the sign is k.

There are many ways to express a sign, but in the case of a circuit code, it is very convenient to use a polynomial. Arbitrary code polynomial c (x) Is expressed as a multiple of the generator polynomial. GF (2 ^p ) Defined in [n, k] Generating Polynomial of RS Code g (x) Message polynomial m (x) If each of

g (x) = g ₀ + g ₁ x + ⋅⋅⋅ + g _nk x ^nk ,

m (x) = m ₀ + m ₁ x + ⋅⋅⋅ + m _k-1 x ^k-1

Information polynomial m (x) on x ^nk Multiply by and generate this polynomial g (x) Divide by the remaining surplus polynomial P (x) The sign polynomial c (x) Is expressed as:

c (x) = x ^nk m (x) + P (x)

When applying RS code in practice q = 2 ^m In most cases, we consider only this case. Error correction capacity is t GF (2 ^m ) The RS code on the top is configured as follows. GF (2 ^m ) Primitive element α Say polynomial g (x) Is 2t consecutive α Have powers of.

Since the degree of the generation polynomial is exactly 2t, let k be the dimension of RS. k = n-2t Becomes

Since the RS code is a non-binary code, not only an error location but also an error value (erroe value) at that location must be determined.

2 is a flowchart showing a decoding method using a conventional RS code. The method shown in FIG. 2 is performed as follows.

Syndrom S _{0 -} S _2t-1 Calculate (S210).

Find the error location (S220).

If the number of errors is t or more, the decoding failure is declared and the decoding ends (S230, S240).

The error value is determined (S250).

After the error is corrected, the decoding is finished. (S260)

Sign polynomial c (x) Polynomial when is transmitted over a channel e (x) Assuming that is added, the received polynomial is r (x) = c (x) + e (x) It is expressed as Generate polynomial for RS codes that can correct t errors g (x) Since the degree of is 2t, the proof consists of 2t elements S _ib = r (α ⁱ ) = e (α ⁱ ), b ≦ ⁱ ≦ (b + 2t-1) Calculate Syndrome polynomial S (z) And ν ( ν≤t Error locator polynomial c (z) , Error evaluator polynomial w (z) Let's define

σ (z) The roots of α ^{-j One} , α ^{-j 2} , ⋅⋅⋅, α ^{-j ν} Each of which is the inverse of the error location. From the above equation, the following key equation can be derived.

σ (z) S (z) = w (z) mod z ^2t

Solve these main equations to find the root of the error location polynomial, determine the location of the error, and estimate the error equation by determining the error value using Forney's equation. Finally, correct the error by adding the estimated error equation to the received polynomial.

That is, the estimated received polynomial silver

to be.

here, Is the estimated error equation.

Finite body GF (2 ^m ) Create polynomials on g (x) Having [2 ^m -1, 2 ^m -2t] The sign polynomial of the sign c (x) Let's say

g (x) = (x-α ^b ) (x-α ^{b + 1} ) ⋅⋅⋅ (x-α ^{b + 2t-2} )

Where b is an integer. By extending this code [2 ^m , 2 ^m -2t] The code is constructed as follows. Sign polynomial Where It is defined as At this time, t becomes the error correction capability of the extended code.

3 shows a codeword configuration of an extended code. [2 ^m , 2 ^m -2t] Sign (sign C ) C ′ Is the sign C And the last symbol part It is made of. Decoding is not possible with the existing decoding method for the RS code extended to the special form as described above, and a new algorithm is needed to decode it.

The present invention has been made to solve the above problems. [2 ^m , 2 ^m -2t] It is an object of the present invention to provide a method for obtaining a false test for decoding an extended Reed-Solomon code.

Another object of the present invention is to use the above [2 ^m , 2 ^m -2t] The present invention provides a method for decoding an extended Reed-Solomon code.

1 is a block diagram showing the configuration of a general digital communication system.

2 is a flowchart illustrating a decoding process for a conventional Reed-Solomon (RS) code.

3 shows the format of an extended RS code.

4 is a flowchart showing a decoding method according to the present invention.

5 shows a case where an error occurs in an extended RS code.

According to the present invention to achieve the above object [2 ^m , 2 ^m -2t] The method for obtaining the false test for decoding Reed-Solomon code is

[2 ^m -1, 2 ^m -2t] Where m is a positive integer and t is the ability to correct errors. c (x) , r (x) , And e (x) , [2 ^m , 2 ^m -2t] Sign polynomial, receive polynomial, and error polynomial of Reed-Solomon code, respectively. (here, ego, α Is a finite body GF (2 ^m ) Primitive element, b is an integer), (here, ), And (here, ),

It is characterized by obtaining the testimony determined by.

According to the present invention to achieve the above other object [2 ^m , 2 ^m -2t] The decoding method of the Reed-Solomon code is

[2 ^m , 2 ^m -2t] Where m is a positive integer and t is the error-correction capability. C ′ In the decoding method of

[2 ^m -1, 2 ^m -2t] Reed-Solomon code C Sign polynomial, receive polynomial, and error polynomial c (x) , r (x) , And e (x) , [2 ^m , 2 ^m -2t] Sign polynomial, receive polynomial, and error polynomial of Reed-Solomon code, respectively. (here, ego, α Is a finite body GF (2 ^m ) Primitive element, b is an integer), (here, ), And (here, ),

(a) code C A misconception about S _{0 -} S _2t-1 Obtaining through the following equation;

(b) S _{0 -} S _2t-2 Using the sign C Obtaining the location of the error for;

(c) if C Error in t-1 If less than the symbol, go to step (g);

(d) code C A misconception about S _2t-1 The process of finding the location of the error using further;

(e) if C Error in t If it occurs below the symbol, go to step (g);

(f) declaring a decoding failure and terminating it;

(g) determine the error value corresponding to the error location, and C Codeword c (x) By estimating To find and Calculating the process;

(here, ego, Is an estimated error equation)

(h) sign C ′ Codeword c ′ (x) It is characterized in that it comprises a process of estimating and decoding. Hereinafter, the configuration and operation of the present invention will be described in detail with reference to the accompanying drawings.

Error polynomial To receive the polynomial Let's define Extended code in this form C ′ In decoding C Take advantage of the misconception. sign C The testimony of appears in a special form and is derived as in Equation 1 below.

4 is a flowchart showing a decoding method according to the present invention. Extended in a special form according to the invention shown in FIG. [2 ^m , 2 ^m -2t] The decoding method for the Reed-Solomon code is as follows.

sign C A misconception about S _{0 -} S _2t-1 Is obtained through Equation 1 (S410).

S _{0 -} S _2t-2 Using the sign C Obtain the location of the error for (S420).

In case C Error in t-1 If less than the symbol, go to S470 (S430).

sign C A misconception about S _2t-1 Use it to find the location of the error (S440).

In case C Error in t If it occurs below the symbol, the flow goes to S470.

Declares the failure of decoding and terminates (S460).

Determine the error value corresponding to the error location, and sign C Codeword c (x) By estimating To find (S470), and (S480)

sign C ′ The error is corrected and the decoding ends. (S490)

The decoding flowchart according to the present invention is shown in FIG.

sign C A testament to S Extended code C ′ A testament to S ′ Let's say There is then a relationship between the two signs of the proof.

For example, the sign C ′ Error correction ability t If it's a symbol t An error below the symbol may occur as illustrated in FIG. 5. Extended sign as can be seen in Figure 5 C ′ In the first symbol 2m-1 Sign up to the first symbol C Part of the 0th symbol You can think of it in parts. In the case of (a) and (b) in FIG. C Error in t-1 Occurs below symbols S _{0 -} S _2t-2 Code using only C Error polynomial in e (x) Can be estimated. Where the codeword c (x) By estimating Obtain The error of the 0th symbol is Becomes

In the case of (c) in Figure 5 C Is more than error correction t False because a symbol error occurred S _2t-1 Using additional signs C Error polynomial in e (x) Estimate At this time, error does not occur at 0th symbol. Becomes therefore, ego,

Becomes

As described above, the decoding method according to the present invention [2 ^m , 2 ^m -2t] It can be implemented in digital systems using extended Reed-Solomon codes. For example, a cable modem based on ITU-U's J.83 Annex B standard m = 7, t = 3 As if [128, 122] An extended Reed-Solomon code is used and can be decoded by the method according to the present invention.

Claims (3)

[2 ^m , 2 ^m -2t] (Where m is a positive integer and t is an error correction capability) in the extended Reed-Solomon code decoding method, [2 ^m -1, 2 ^m -2t] Sign polynomial, receive polynomial, and error polynomial of Reed-Solomon code, respectively. c (x) , r (x) , And e (x) , [2 ^m , 2 ^m -2t] Sign polynomial, receive polynomial, and error polynomial of Reed-Solomon code, respectively. (here, ego, α Is a finite body GF (2 ^m ) Primitive element, b is an integer), (here, ), And (here, ), The decoding method, characterized in that to obtain a miscalculation determined. [2 ^m , 2 ^m -2t] (Where m is a positive integer and t is error correction capability) extended Reed-Solomon code C ′ In the decoding method of [2 ^m -1, 2 ^m -2t] Reed-Solomon code C Sign polynomial, receive polynomial, and error polynomial c (x) , r (x) , And e (x) , [2 ^m , 2 ^m -2t] Sign polynomial, receive polynomial, and error polynomial of Reed-Solomon code, respectively. (here, ego, α Is a finite body GF (2 ^m ) Primitive element, b is an integer), (here, ), And (here, ), (a) code C A misconception about S _{0 -} S _2t-1 Obtaining through the following equation; (b) S _{0 -} S _2t-2 Using the sign C Obtaining the location of the error for; (c) if C Error in t-1 If less than the symbol, go to step (g); (d) code C A misconception about S _2t-1 The process of finding the location of the error using further; (e) if C Error in t If it occurs below the symbol, go to step (g); (f) declaring a decoding failure and terminating it; (g) determine the error value corresponding to the error location, and C Codeword c (x) By estimating To find and Calculating the process; (here, ego, Is an estimated error equation) (h) sign C ′ Codeword c ′ (x) Estimating and finishing decoding. [2 ^m , 2 ^m -2t] (Where m is a positive integer and t is an error correction capability) in the extended Reed-Solomon code decoding method, [2 ^m -1, 2 ^m -2t] Testimony of the Reed-Solomon Code S _{0 -} S _2t-1 and [2 ^m , 2 ^m -2t] Testimony of the Extended Reed-Solomon Code S ' _0- S' _2t-1 Decryption using the relationship between.