KR100258440B1 - Decoding algorithm for shortened bch code - Google Patents

Abstract

PURPOSE: A decoding algorithm for shortened BCH code is provided to correct an error if the error occurs within the shortened distance of the BCH code depending on the condition regardless of the number of errors occurred. CONSTITUTION: In a decoding algorithm for shortened BCH code, a transmission data is sent through a channel(32) after the data is encoded by a BCH encoder(31). The BCH codeword received goes through a fourier transform by a fourier transform calculator(33). An EFIA(Extended Fundamental Iterative Algorithm) calculator(36) calculates the syndrome obtained by the calculator(33). Another syndrome calculator calculates the other syndrome. All the calculated syndrome are produced as an error polynomial expression through a reverse fourier transform. A decoding detector finding the error position(38) determines whether the error has occurred within the shortened distance with the error polynomial expression.

Description

Decoding algorithm for shorter BCH code

The present invention relates to a decoder for error correction protection, and more particularly, to a decoding method of a shortened BCH code and a decoder suitable for the same.

The BCH code is a typical error correction code that can be encoded and decoded using algebraic properties, and is widely used because it is relatively easy to implement an encoder and a decoder. However, decoders of the BCH code which are widely used now can correct only errors due to the BCH limit rather than the actual error correction capability of the code. On the other hand, in recent years, methods for correcting errors up to the actual error correction capability of the code, not the BCH limit, have been developed.

In this BCH code, the input and output lengths of the block code are strictly limited by the field implementing the code or the error correction capability. However, it is difficult to use error correction codes with fixed input and output lengths because there are limitations on the frame size and code rate in some systems. Therefore, most systems do not use a portion of the input and a portion of the output, so that a short code that satisfies the constraints on the frame size and the code rate is used.

In the case of using an error correction code, output data having a parity bit attached to the input data is generated, so that the distance difference between the encoded output data increases than the distance difference between the input data. When the transmitting end transmits the data encoded using the error correction code, the receiving end compares the received data with the error and the data that can be output when the error correcting code is used, and determines that the outputable data having the smallest difference is the data originally transmitted. . Therefore, the minimum difference between the output data of the error correction code is closely related to the error correction capability.

If the input data of the BCH code is composed of q-ary symbols, the length of the input is n = q ^m -1, and the required error correction capability is t, the generation polynomial g (x) of the BCH code is Same as

Where f _i (x) is the minimal polynomial of a ⁱ , the i-multiplier of a primitive a of GF (q).

The generated polynomial g (x) satisfies the condition of Equation 2 below.

When the input data is expressed by the polynomial i (x) in GF (q), Equation 3 is expressed.

The coding of the BCH code is represented by the product of the input polynomial i (x) and the generated polynomial g (x). The output polynomial c (x) can be represented by the following equation (4).

c (x) = i (x) g (x)

If the length of the output data is n, the order of the output polynomial c (x) is n-1. The order of the generated polynomial g (x) is determined by Equation 1. Therefore, the order k-1 of the input polynomial i (x) is also determined, and the BCH code defined above becomes the (n, k, t) code.

Equation 4 may be modified to represent the same BCH code as Equation 5.

는 a(x)를 g(x)로 나눈 나머지를 의미한다.here,

Is the remainder of a (x) divided by g (x).

Equations 4 and 5 represent a method of encoding BCH codes in both equations (n, k, t), and Equation 4 is referred to as an unstructured code and Equation 5 is referred to as an organizational code.

In the organization type code, only parity bits are attached to the output data. Therefore, the input data appears intact in the output data. Therefore, not using k ₀ input data has the same effect as not using k ₀ output data. Such a code is referred to as (nk ₀ , kk ₀ , t) shortened BCH code. The output polynomial c (x) satisfies the same properties as in Equation 6 by Equation 2.

Even in the case of the organization code, R _{g (x)} [c (x)] = 0, thus satisfying c (a ^j ) = 0, j = b, b + 1, ..., b + 2t-1.

If the transmitting end encodes the transmission data and then transmits the data, the BCH encoded data may receive incorrect data when an error occurs during the channel. In this case, if the channel existing between the transmitting end and the receiving end is an addition channel, the received data may be expressed as shown in Equation (7).

v (x) = c (x) + e (x)

Here, v (x) is a polynomial expression of the received data, and e (x) is a polynomial expression of an error occurring in the channel.

According to Equation 6, the following Equation 8 can be obtained.

Here, e (a ^j ) is called syndrome S _j . The number of syndromes that can be obtained by Equation 8 is 2t, and the decoding of the BCH code can be performed by obtaining all syndromes. Therefore, the core of the decoding algorithm of the BCH code is a method of calculating other syndromes with 2t syndromes.

Hereinafter, the decoding method of the BCH code will be described in detail with reference to the drawings.

1 is a block diagram for explaining a decoder of a BCH code using a conventional free transform.

The transmission data is transmitted through the channel 12 after being encoded by the encoder 11 of the BCH code. The receiving end receives the data transmitted through the channel 12 and then pre-transforms the received data with a Fourier Transform calculator 13.

In this case, the Fourier transform in the finite field is defined as in Equation 9 below.

The 2t values of the pre-converted data are equal to V _j = S _j , j = b, b + 1, ..., b + 2t-1. With the syndromes of 2tro calculated here, the remaining syndromes are calculated using the syndrome calculator 16. When the number of errors occurring in the channel 12 is less than or equal to t, syndromes are converted into free values of the error polynomial e (x) as shown in Equation 8. Transform the output of the syndrome calculator 16 into an inverse freer with an Inverse Fourier Transform calculator 14 to obtain the error polynomial e (x). The inverse Free Four transformation of the syndrome calculator 16 output is given by the following equation (10).

The subtractor 15 subtracts the error value that is the output of the inverse Fourier transform calculator 14 from the BCH encoded data received from the channel 12 to complete decoding of the BCH code. The decoder shown in FIG. 1 is limited to a BCH threshold value smaller than 1/2 of the minimum distance, thereby enabling error correction.

2 is a block diagram for explaining a decoder of a BCH code published by the conventional Feng-Tzeng.

The decoder of the BCH code shown in FIG. 2 is the same except that the syndrome calculator 16 of FIG. 1 is separated into an Extended Fundamental Iterative Algorithm (EFIA) calculator 26 and the remaining syndrome calculator 27. In order to calculate the syndromes, the syndromes are calculated by the EFIA calculator 26 with the syndrome values output from the pre-transformation calculator 23. The remaining syndromes that are not calculated by the remaining syndrome calculator 27 and the EFIA calculator 26 are calculated using the syndrome value output from the EFIA 26. The inverse Fourier transform calculator 28 calculates an error polynomial by introducing values obtained by calculating all syndromes output from the remaining syndrome calculator 27.

Hereinafter, the operation principle of the EFIA calculator 26 and the remaining syndrome calculator 27 will be described in detail. Equation 11 of the matrix S may be constructed from syndromes.

Where S _{i, j} = S _{b + i + j-2} . Among the elements of the matrix S, S _t to S _{b + 2t-1} are the values that can be obtained by pre-converting the received data, and the remaining syndromes cannot be calculated.

If the number of errors occurring in the channel is υ, the rank of the syndrome matrix S implemented as described above becomes υ when υ ≤ t. Therefore, the fact that the rank of the syndrome matrix S is less than or equal to t can be used to calculate unknown syndromes.

The FIA (Fundamental Iterative Algorithm) is a minimum l search algorithm in which M × L matrices are dependent on each other in the M × N matrix, and Feng-Tzeng transforms the FIA to use it for decoding the BCH code. Extended Fundamental Iterative Algorithm. The operation of this EFIA is performed in the following steps.

Step S1: Sets tables C, D, E, and F and substitutes 1 → s (order), 1 → r (r-th column), and 1 → c ^{(o, s)} , respectively.

를 산출한다.S2 stage:

To calculate.

Step S3: If the calculated values of d _{r and s} are 0, the process proceeds according to the conditions of steps S3 (a) and S3 (b).

Step S3: r is not equal to the value of 2t + 1-s, and proceeds according to the conditions of steps S3 (a1) and S3 (a2).

로 두고 S_r+1.s와 c^{(r-1, s)}(x)를 각각 테이블 E와 테이블 F에 저장한 다음 S3(a2)단계로 진행한다.Step S3 (a1): If d _{r + 1, μ} does not exist in the table D, S _{r + 1, s} =

S _{r + 1.s} and c ^{(r-1, s)} (x) are stored in Table E and Table F, respectively, and then proceed to step S3 (a2).

Step S3 (a2): If d _{r + 1.μ} exists in table D and s is equal to 2t, the _algorithm ends the rhythm. If s does not equal 2t, then c ^{(r-1, s)} (x) → c ^(s) (x) → c ^{(0, s + 1)} (x). Then substitute s + 1 → s and 1 → r and proceed to step S2.

S3 (b) step: r is not equal to 2t + 1-s, substitute c ^{(r-1, s)} (x) → c ^{(r, s)} (x), r + 1 → r and go to step S2 Proceed.

Step S4: If d _{r and s} are not 0, the process proceeds according to the conditions of steps S4 (a) and S4 (b).

로 대입하고 S3(a)단계로 진행한다.Step S4 (a): If d _{r and μ} exist in the table D in 1≤μ <s,

Substitute at and proceed to step S3 (a).

S4 (b) a step 1≤μ <s at d _{r, μ} does not exist in the table D, and stores the d _{r, μ} in Table D, and stores the c ^{(s) (x)} in Table C. Then proceed to step S3 (a2).

이다. 그리고

의 s번째 항의 계수를 의미한다.here,

to be. And

The coefficient of the sth term of.

When the EFIA is completed, predetermined syndromes are calculated and stored in table E, and polynomial c (x) for storing syndromes in table F is stored. The EFIA calculator 26 of FIG. 2 implements EFIA.

Not all syndromes are calculated by the EFIA calculator 26, but using the syndromes calculated by the pre-transformation, the syndromes calculated by EFIA stored in table E, and the polynomial c (x) stored in table F. The remaining syndromes are calculated such that c (x) multiplies the previous syndromes by zero. The remaining syndrome calculator 27 performs the remaining syndrome calculation as described above. The value output from the remaining syndrome calculator 27 is inverse pre-transformed by the inverse free transform calculator 24 to generate an error polynomial, and the error value of the generated error polynomial is received through the channel 12 in the subtractor 25. The decoding is completed by subtracting from the BCH encoded data. In this case, decoding is possible up to 1/2 of the minimum distance improved from the decoder shown in FIG. 1, that is, to the actual distance.

However, since the button BCH code does not use a portion of the input in comparison with the original BCH code, the set of outputable data of the shortcode becomes a subset of the set of outputable data of the original BCH code. That is, the minimum difference between the outputable data of the shortcode is greater than or equal to the minimum distance between the outputable data of the original code, which means that the shortened BCH code will have a greater error correction capability compared to the original BCH code. The decoder has been required to have an improved error correction function.

The technical problem of the present invention relates to a method of decoding BCH encoded data and a decoder suitable for the error, when an error cannot be corrected by a conventional BCH code decoder when a shortened BCH code is used. .

1 is a block diagram for explaining a general decoder of a BCH code using a conventional free transform.

2 is a block diagram of a decoder capable of decoding up to the actual distance of the BCH code published by the conventional Feng-Tzeng.

3 is a block diagram for explaining a decoder of a shortened BCH code according to the present invention.

* Explanation of symbols for main parts of the drawings

31: BCH encoder 32: channel

33: Free Conversion Calculator 34: Inverse Free Conversion Calculator

35: Subtractor 36: EFIA Calculator

37: remaining syndrome calculator 38: decoding error position detector

39: buffer 40: decoding controller

In the BCH decoding method for pre-transforming and decoding shortened BCH-coded data according to the present invention for achieving the above object, a syndrome is calculated through an EFIA (Extende Fundamental Iterative Algorithm) with a syndrome calculated by pre-transforming. A first process of calculating the remaining uncalculated syndrome, a second process of calculating the error polynomial by converting the calculated syndrome into an inverse free, and not having an error in a shortened position of the BCH-coded data through the error polynomial In the third step of subtracting the error polynomial value from the BCH-coded data, if it is confirmed, predetermined values are sequentially assigned to elements that are likely to be syndromes if an error exists in a shortened position through the error polynomial. In combination with the first and second processes to calculate the syndrome and the error polynomial If there is one error polynomial in which the error is not in the shortened position among the error polynomials calculated in the fourth step, and the error polynomial of the error polynomial is subtracted, And a fifth process of decoding.

In the fourth process, the element likely to be the syndrome is an element in a column of S _{1,2t + 1} and S _{1,2t + 2} in the syndrome matrix S of GF (q ^m ). .

Where t is the actual distance of the BCH code.

Further, in the fourth process, predetermined values of the element may be values ranging from 0 to q ^m −1.

In addition, in the BCH decoder which decodes the shortened BCH-coded data through the pre-transformation calculator according to the present invention for achieving the above object,

A syndrome calculator that calculates syndromes calculated by the preier conversion calculator, introduces predetermined values into elements likely to be syndromes, and calculates syndromes, and an inverse freeier transform that converts the calculated syndromes to an inverse frie and outputs an error polynomial. Calculator, Decoded error position detector that checks whether an error occurs in the shortened position by introducing the error polynomial and outputs an error position detection signal, and stores and outputs only the error polynomial in which no error occurred at the shortened position according to the light control signal. A buffer for inputting a control signal and outputting the stored error polynomial, a subtractor for subtracting and decoding the error value of the error polynomial outputted from the buffer from the BCH-coded data, and a position where the error is shortened through the error position signal If it is found in, it could be a syndrome And sequentially output predetermined values, output the write control signal if no error exists in the shortened position, and after the calculation of the error polynomial for one stream of the BCH-coded data is completed, the buffer is And a decoding controller for outputting the output control signal when the error polynomial searched and stored is one.

In addition, the syndrome calculator is characterized in that it comprises an Extended Fundamental Iterative Algorithm (EFIA) calculator and the remaining syndrome calculator for calculating the syndrome not calculated in the EFIA calculator.

3 is a block diagram illustrating a decoder of a shortened BCH code according to the present invention.

The front end of the decoder of the shortened BCH code according to the present invention of FIG. 3 operates the same as the front end of the decoder shown in FIG. Therefore, the transmission data is encoded by the BCH encoder 31 and then transmitted through the channel 32, and the BCH code received at the receiver is pre-transformed through the pre-transformation calculator 33, and 2t syndromes obtained by the pre-transformation are obtained. Have them. Where t is the error correction capability and represents the actual distance as one half of the minimum distance. After calculating the syndromes with the EFIA calculator 36 with 2t syndromes, all syndromes are calculated with the remaining syndrome calculator 37. All the calculated syndromes are computed as error polynomials through the inverse freer transform calculator 34. The decoding error position detector 38 has an error polynomial obtained by the pre-transformation calculator 34 and detects whether the decoding error has occurred at a shortened or non-shortened position.

If more errors than the error correction capability t occur in the channel 32, the decoder generates a decoding error by determining that an error has occurred at a place different from where the actual error has occurred, or incorrectly determining that no error has occurred. At this time, when a shortened BCH code is used, there is a case where an error has occurred in an input symbol shortened by a decoding error. If this happens, it can be determined that a decoding error has occurred.

The decoding error position detector 38 checks whether an error has occurred at the shortened position by introducing the error polynomial calculated through the inverse free conversion calculator 34 and outputs an error position signal to the decoding controller 40. The decoding controller 40 outputs a write signal to the buffer 39 and outputs a write signal to the buffer 39 if it is recognized that all the generated errors are not present in the shortened position through the error position detection signal. After storing the polynomial in the buffer 39 and outputting it to the subtractor 35, the subtractor 35 subtracts an error value of the decoded polynomial from the introduced BCH coded data to decode the BCH coded data of one stream. To complete.

On the other hand, if the decoding error position detector 39 detects that there is a decoding error in the shortened position where the decoding error cannot occur through the error polynomial, and outputs the error position signal to the decoding controller 40, the decoding controller 40 is a syndrome. The syndrome value of this possible element is added and sent to the EFIA calculator 36. The EFIA calculator 36 adds two columns having elements S _{1, 2t + 2} and S _{1, 2t + 3} in Equation 11, and an element of GF (q ^m ) that is likely to be a syndrome. Two of the elements S _{1, 2t + 1,} and S _{1, 2t + 2} are set as the values introduced from the decoding controller 40 to calculate the syndrome so that t + 1 errors can be corrected. The error polynomial is calculated using the remaining syndrome calculator 37 and the inverse freer transform calculator 34. The decoding error position detector 38 checks whether an error has occurred at the shortened position with the error polynomial obtained by the inverse Fourier transform calculator 34. If it is determined that all the errors generated through the error position signal are not present in the shortened position, the decoding controller 40 outputs the write control signal to the buffer 39 to output the error polynomial which is the output of the inverse free conversion calculator 34. Save it. The above series of operations are performed for all pairs with symbol values of 0 to q ^m which are likely to be syndromes. When the output of the error polynomials for all symbol values is completed, the decoding controller 40 searches for how many error polynomials are stored in the buffer 39, and if there is one stored error polynomial, the decoding controller 40 can correct errors. Therefore, the output control signal is transmitted to the buffer 39 to output the stored error polynomial to the subtractor 35. The checker 35 completes decoding by subtracting the error value of the error polynomial from the received BCH encoded data. If there are two or more error polynomials stored in the buffer 39, since error correction of the received BCH code is impossible, the received BCH encoded data is output as it is.

As described above, the present invention can decode up to the actual distance of the BCH code, and in the case of the shortened BCH code, even if more errors occur than the conventional BCH code error correction capability, if an error occurs at a shortened position of the BCH code, Because error correction is possible, more advanced decoding function is provided.

Claims (7)

A BCH decoding method for pre-transforming and decoding shortened BCH-coded data, comprising: a first process of calculating a syndrome through EFIA (Extended Fundamental Iterative Algorithm) with a syndrome calculated by pre-transforming and calculating the remaining uncalculated syndrome; A second step of calculating the error polynomial by converting the calculated syndrome to inverse free, and if the error does not exist at a shortened position of the BCH coded data through the error polynomial, the error is input from the BCH coded data. The third process of subtracting and decoding the value of the polynomial, through the first process and the second process by sequentially combining the predetermined values to the element that is likely to be a syndrome if an error exists in the short axis position through the error polynomial In a fourth process of calculating a syndrome and calculating error polynomials, and in the fourth process Shortened BCH comprising the fifth step of subtracting and decoding the BCH coded data introduced into the error value of the error polynomial if one error polynomial among the error polynomials is present in the shortened position. Decoding method of code. The element of claim 1, wherein the element likely to be the syndrome in the fourth process is an element in a column of S _{1,2t + 1} and S _{1,2r + 2} in the syndrome matrix S of GF (q ^m ). Decoding method of a shortened BCH code characterized in that. (Where t is the actual distance of the BCH code) 2. The method of claim 1, wherein the predetermined values of the element range from 0 to q ^m −1. In the BCH decoder which decodes the shortened BCH-coded data through the Preier Transformation Calculator, a syndrome calculator for calculating syndromes calculated by the Preier Transformation Calculator and injecting predetermined values into elements likely to be syndromes to calculate the syndromes. A reverse error free conversion calculator for converting the calculated syndrome to an inverse free output and outputting an error polynomial; a decoded error location detector and a light control signal for outputting an error location detection signal by checking whether an error occurred by shortening the error polynomial; A buffer for storing only the error polynomial in which the error does not occur at the shortened position and outputting an output control signal to output the stored error polynomial, and subtracting and decoding the error value of the error polynomial output from the buffer from the BCH encoded data. The subtractor and the error position signal through the error If it is confirmed that it exists in the shortened position, the predetermined value of the element is sequentially combined and output to the element which is likely to be a syndrome, and if the error does not exist in the shortened position, the light control signal is output, and the BCH encoding is performed. And a decoding controller for searching the buffer and outputting the output control signal if the stored error polynomial is one after the calculation of the error polynomial for one stream of the data is completed. 5. The decoder of claim 4, wherein the syndrome calculator comprises an Extended Fundamental Iterative Algorithm (EFIA) calculator and a remaining syndrome calculator for calculating syndromes not calculated in the EFIA calculator. 5. The short axis according to claim 4, wherein the element likely to be a syndrome is an element in a column of S ^{1,2r + 1} and S ^{1,2r + 2} in the syndrome matrix S of GF (q ^m ). Decoder for BCH code. (Where t is the actual distance of the BCH code) 5. The decoder of claim 4, wherein the predetermined values of the element range from 0 to q ^m −1.

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