RU2340088C2 - Syndrome decoding method of decoding recurrent code (versions) - Google Patents

Abstract

FIELD: information technology.

SUBSTANCE: for the received distorted coded implementation, an extended syndrome is calculated, and, using an extended test matrix, the sets of lines, which are linear combinations of the calculated syndrome, are determined. Each linear combination expressly corresponds to an error vector and for each error vector, a metric is calculated, and for correcting the most probable errors, the correction vector during decoding with a "solid" solution, an error vector is chosen, which has minimum metric (minimum weight error vector), and during decoding with a "soft" solution, an error vector is chosen, with maximum modified metric. Elements are then inverted in the systematic part of the code word, the numbers of which corresponds to the positions of non-zero elements of the correction vector.

EFFECT: increased authenticity of decoding and reduced device and computational complexity.

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Description

The invention relates to the field of communication technology, in particular to information transmission systems in which cyclic linear block codes are used to protect it from distortion in the communication channel, and can be used in codecs (codecs) of data transmission systems, as well as in noise-immune devices encoding during transmission and storage of discrete information.

A known method of decoding linear block (n, k) codes [1-3] (n is the codeword length, k is the number of encoded bits or the systematic part of the codeword), which includes the calculation of the (nk) -bit codeword syndrome s distorted in the communication channel, the choice of the correction vector that corresponds to the syndrome s from a pre-formed table of leaders of adjacent classes of error vectors, the bitwise addition of the correction vector by mod 2 to the distorted codeword, and the allocation of the corrected systematic part of the codeword.

The disadvantages of this decoding method include the need to store a leader table of adjacent classes of error vectors, the volume of which is 2 ^nk × n-bit (or k-bit) vectors, which is technically impossible for large values of nk.

The aim of the invention is to increase the reliability (quality) of decoding, as well as reducing hardware and computational complexity and, in particular, implementing the optimal decoding procedure with both a “hard” and a “soft” solution.

To achieve the goal, a method for decoding a cyclic linear block (n, k) -code is proposed, in which the correction vector e is not selected from the leader table of adjacent classes of error vectors, but is formed as a result of representing the n-bit extended syndrome s in the form of linear combinations of lines of the extended check matrices H _{n, n} . The line numbers of the extended check matrix unambiguously indicate the numbers of the distorted positions in the received distorted codeword. The expanded verification matrix H _{n, n} is the transposed verification matrix [1], in which the jth row consists of coefficients for an unknown polynomial:

h _j (x) = x ^j h (x) mod (x ⁿ +1),

where h (x) = (x ⁿ +1) / p (x) is the verification polynomial of the cyclic (n, k) -code,

p (x) is the generating code polynomial,

j = 0, 1, ... n-1.

The transposed extended verification matrix H _{n, n} differs from the verification matrix H _{n, nk} [1, 2] only in that it adds k more rows, each of which is also a cyclic shift of the previous row by one digit to the right.

To decode a cyclic linear block (n, k) code, a method is proposed, which consists in the following.

(h_i - коэффициенты при неизвестном проверочного многочлена h(x) циклического линейного блочного (n,k)-кода).For the received distorted code implementation (b⊕e _ks ), the extended syndrome s = (b⊕e _ks ) × Н _{n, n} is calculated, in which the position j of any nonzero element is determined. Then, in the expanded verification matrix H _{n, n} , those rows are found that contain also a nonzero element in the jth position. Next, the current syndromes s _{1, i of the} first iteration step are calculated (the first step of the iterative procedure for representing the expanded syndrome in the form of linear combinations of rows of the expanded verification matrix H _{n, n} ). To do this, each of the found lines is added mod 2 with expanded s syndrome. Each current syndrome s _{1, i} is associated with a vector N _{1, i} , in which the row number of the expanded check matrix H _{n, n} , which participated in the formation of the current syndrome s _{1, i} , is entered. Such pairs of vectors s _{1, i} , N _{1, i} at the first step of the iteration will be formed

(h _i are the coefficients of the unknown verification polynomial h (x) of the cyclic linear block (n, k) code).

If one of the current syndromes s _{1, i} is a zero vector, then the minimum weight error vector is found, and the codeword element distorted in the communication channel is indicated in N _{1, i} . After correction (inversion) of the distorted element, the corrected systematic part of the codeword is selected as decoded information. This decoding ends.

If none of the w current syndromes s _{1, i} is a zero vector, then go to the next step of the iteration. For each current syndrome of the previous mth iteration step s _{m, i,} w pairs of vectors s _{m + 1, i} , N _{m + 1, i} are calculated, as well as in the first step of the iteration. The total number of pairs of vectors s _{m + 1, i} and N _{m + 1, i} at the m + 1st step is w _{m + 1} = w ^{m + 1} .

If at least one of the current syndromes s _{m + 1, i} is the zero vector, then the minimum weight error vector is found, and the codeword elements distorted in the communication channel are indicated in N _{m + 1, i} . After correction (inversion) of the distorted elements, the corrected systematic part of the codeword is selected as decoded information. This decoding ends.

If none of w _{m + 1} current syndromes s _{m + 1, i} is a zero vector, then go to the next step of iteration, etc., until the current syndrome, which is a zero vector, is generated. Then, after correction (inversion) of the distorted elements, the corrected systematic part of the codeword is selected as decoded information.

When decoding perfect codes with a “hard” solution, searching for the optimal correction vector (minimum weight vector) will require no more than t iterations (where t is the multiplicity of corrected errors (n, k) code).

In the case when imperfect codes with a “hard” decision are decoded, a variant of the method described above is proposed, for which it is also necessary to perform no more than t iterations. If a syndrome that is a zero vector is not found at the t-th iteration step, then it is necessary to abandon the correction of this distorted codeword, since the weight of the correction vector exceeds the multiplicity of the corrected errors of the (n, k) code and is equal to t + 1. If, nevertheless, it is necessary to correct the correctable t + 1-fold errors of the imperfect code, then it is necessary to perform t + 1 iteration. Moreover, if at the t + 1st iteration among the calculated syndromes only one is the zero vector, then a correctable t + 1-fold error is found, the distorted elements of which are given by the vector N _{t + 1, i} . If several syndromes are zero vectors at the t + 1st iteration among the calculated syndromes, then an uncorrectable t + 1-fold error is found and correction should be abandoned.

In the case when an “extended” code [1-3] with a “hard” decision is decoded, variants of the methods described above are proposed in which, before decoding, an extension element is removed from a possibly distorted code word obtained through a communication channel.

In the case where a “shortened” code [1-3] with a “hard” solution is decoded, variants of the methods described above are proposed, in which, before decoding, zero elements are added to the beginning of the possibly distorted code word, the number of which is equal to shortening. Obviously, the zero positions added to the beginning cannot be distorted in the communication channel, since they were not transmitted through it. This fact allows us to reduce the number of current syndromes S _{m, i} at the m-th iteration, which will reduce the computational complexity of decoding.

In the case when the decoding is performed with a “soft” solution of perfect, imperfect and extended codes, variants of the methods described above are proposed in which for each pair of vectors s _{m, i} and N _{m, i, a} modified metric is calculated:

or

where p (y _j / b _{j, c} ) is the conditional probability that the value of _{j is} obtained at the output of the demodulator when the code word of the cth character is transmitted over the communication channel at the jth position, which can take two values for a binary code: b _{j, 0} = 0 and b _{j, 1} = 1,

e _{i, j} is the j-th element of the i-th error vector e _i belonging to the adjacent class of errors of the syndrome s _{m, i} ,

The modified metric (1) and (2) is the ratio of the generally accepted metric [1-3] of the i-th code word to the metric received from the communication channel with the “hard” solution of the distorted code word. Correct those pleasant elements with a “hard” solution of the distorted codeword that are given by the vector N _{m, i} , and for which s _{m, i} = 0, and the metric m _{m, i is} maximum.

In the case where a “shortened” code with a “soft” solution is decoded, a variant of the method described above is proposed in which, before decoding, zero elements with conditional probabilities p (y _j / b _j are added to the beginning of the possibly distorted code word _{, 0} ) = 1 and p (y _j / b _{j, 1} ) = 0.

To reduce computational complexity, variants of the methods described above are proposed in which current syndromes s _{m, i} are excluded from consideration, in which repetition of the positions of distorted elements appears in the vector N _{m, i} .

To reduce computational complexity, variants of the methods described above are proposed in which repeating current syndromes s _{m, i} are excluded from consideration.

The proposed technical solutions allow us to solve the problem of optimal decoding of cyclic linear block codes with both “hard” and “soft” solutions with less computational and hardware complexity than the known decoders [1-3].

The invention is illustrated in the drawing, which shows a structural diagram of an optimal syndromic decoder of a cyclic linear block code.

The cyclic linear block code Syndrome Decoder comprises: a syndrome 20 calculation unit, a current information storage and update unit 40, a read unit 60, a non-zero element search unit 80, an advanced check matrix string search unit 100, a current information calculation unit 120, a distorted codeword correction unit 140, a decoded information generating unit 160.

Syndrome calculating unit 20 for calculating the syndrome s extended by the formula s = (b⊕e _kc) × H _{n, n.}

The block for storing and updating current information 40 is designed to store the current values of the syndromes s _{m, i} and the corresponding vectors N _{m, i} , as well as when decoding with a “soft” solution of the metrics v _{m, i} . The block for storing and updating current information 40 is a random access memory device that can be implemented as a stack (“first come, first serve”), in which each row consists of two (three) fields for storing current vectors s _{m, i} , N _{m, i} , and when decoding with a “soft” solution of the metric v _{m, i} .

The reading unit 60 is designed for sequentially reading the current vectors s _{m, i} , N _{m, i} stored in the storage unit and updating the current information, and when decoding with a “soft” solution, the metrics v _{m, i} . After reading the line, s _{m, i} is transferred to the search unit of the nonzero element 80, as _{m, i} , N _{m, i} and v _{m, i} to the current information calculation unit 120. The read unit 60 can be implemented as a two-way demultiplexer.

The search block for non-zero element 80 is designed to determine the position j of any non-zero element of the expanded syndrome s _{m, i} . It can be implemented in the form of a shift register, in which the leading digit is analyzed for the presence of 0. In this case, the current syndrome s _{m, i is} entered in the shift register and its shifts are performed towards the senior digit until a nonzero value appears in the senior digit element. The number of shifts j, which is the position number of one of the nonzero elements, is fixed by the counter.

The row search block of the extended check matrix 100 is designed to determine the row numbers of the extended check matrix H _{n, n} , for which nonzero elements are located at position j. Block 100 can be made in the form of a shift register, in which the leading digit for the presence of 0 is analyzed, and line number counters, the number of which is equal to w (the number of nonzero coefficients of the verification polynomial h (x)). The j-th column of the expanded verification matrix H _{n, n is} entered into the shift register, which is shifted towards the higher order until it is reset to zero. Line number counters count the number of shifts and when a nonzero element appears in the high order of the shift register, they are sequentially disabled. When the shift register is reset to zero, the numbers of the required rows of the extended verification matrix are in the counters.

The current information calculation unit 120 is intended to calculate current syndromes s _{m + 1, j} = s _{m, i} ⊕ h _j (where h _j is one of the w rows of the extended verification matrix H _{n, n} , the numbers of which are received via bus 110). Then it forms for each current syndrome s _{m + 1, j the} vector N _{m + 1, j} by adding the row number h _j to the vector N _{m, i of the} extended verification matrix H _{n, n} . When decoding with a “soft” solution using formulas 1 or 2, metrics v _{m + 1, j} corresponding to the syndromes s _{m + 1, j} are calculated.

Correction block 140 performs the inversion of those elements of the received distorted codeword that are indicated in the vector N _{m, i} received via bus 131.

The unit for generating decoded information 160 is intended to highlight the systematic part in the adjusted codeword, which is the decoded information.

The proposed device operates as follows.

The received, possibly distorted in the communication channel, code word (b⊕e _ks ) is received from the output of the communication channel (demodulator) via bus 11 to the correction block of the distorted code word 140, and via bus 10 to the block for computing syndrome 20.

If the syndrome s = 0 calculated in the block for computing the syndrome 20 syndrome (zero vector), then the code word is not distorted and information about it is transmitted via the bus 31 to the correction block of the distorted code word 140. After that, the code word is sent to the block 150 for generating decoded information 160 , in which the systematic part of the codeword is highlighted, and via the bus 170 it enters the output of the decoder. This completes the process of decoding the code word.

If the calculated syndrome s ≠ 0, then via bus 30 it enters the current information storage and updating unit 40. The fields of the vector are N _{m, i} , and when decoding with a “soft” solution, the metric field v _{m, i is} reset.

As soon as the first syndrome s enters block 40, a read block 60 starts immediately, which receives the next syndrome s _{m, i} along the bus 50 along with the vector N _{m, i} and when decoding with a “soft” solution and with the metric v _{m, i} . Further, through the bus 70, the syndrome s _{m, i} , is transmitted to the search unit of the nonzero element 80, and on the bus 71, the syndrome s _{m, i} , the vector N _{m, i} , and when decoding with a “soft” solution, the metric v _{m, i} , is transmitted to a current information calculation unit 120.

Then, the search block of the non-zero element 80 determines the position of any non-zero element j of the extended syndrome and transmits the value of this position via the bus 90 to the string search block of the extended check matrix 100.

Next, the row search unit of the extended verification matrix 100 by the obtained number j determines the row numbers of the extended verification matrix H _{n, n} , for which a nonzero element is located at position j and transmits them via bus 110 to the current information calculation unit 120.

After that, in the current information calculation unit 120, the current syndromes S _{m + 1, j} and the corresponding vectors N _{m + 1, j are formed} , and when decoding with a “soft” solution, the corresponding metrics v _{m + 1, j} are calculated using formulas 1 or 2 .

If one of the current syndromes is s _{m + 1, j} = 0, then when decoding with a “hard” decision on the bus 131, the vector N _{m + 1, j} is transmitted to the correction block 140. The distorted code word received in block 140 via bus 11 is corrected (those elements of the distorted code word that are indicated in the vector N _{m + 1, j are} inverted) and transmitted via bus 150 to the block for generating decoded information 160, in which the decoded information is extracted and on the bus 170 it goes to the output of the decoder. The decoding process is completed. When decoding with a “soft” solution, the vector N _{m + 1, j} is transmitted via bus 131 to the correction block 140 only when the metric v _{m + 1, j is} maximum. Otherwise, s _{m + 1, j} = 0, N _{m + 1, j} and v _{m + 1, j} are sent via bus 130 to the current information storage and update unit 40, and the current information calculation unit 120 proceeds to the calculation of the next syndrome.

If all the calculated syndromes are s _{m + 1, j} ≠ 0, then via bus 130 they are stored with their vectors N _{m + 1, j} and metrics v _{m + 1, j} when decoding with a “soft” solution to the storage unit and updating current information 40.

Then, as previously discussed, the reader 60 on the bus 50 again reads the next extended syndrome s _{m, i} together with the vector N _{m, i} and the metric v _{m, j} when decoding with a “soft” solution, etc. until the current syndrome, which is a zero vector, is calculated. Then, after the correction (inversion) of the distorted elements of the codeword, its corrected systematic part is selected as decoded information, and the decoding process ends.

When decoding “shortened” codes, another design option is to modify the syndrome 20 computation unit, which, before calculating the extended syndrome s, adds zero elements to the beginning of the possibly distorted codeword, the number of which is equal to the amount of code shortening. In addition, the row highlighting unit of the expanded check matrix 100 deletes those rows whose numbers are less than the shortening amount. This design solution allows not only to decode “shortened” linear cyclic block codes, but also to increase the speed of the decoder.

To decode the “extended” codes, another design option is to modify the syndrome 20 calculation unit, which before calculating the extended syndrome s removes the last element (extension element or parity element of the code word) received from the communication channel of the possibly distorted code word. When decoding with a “hard” solution, this element is not taken into account in any of the blocks. When decoding with a “soft” solution, the conditional probabilities of the extension element are used in calculating the metrics in the current information calculation unit 120. This design solution allows you to decode the “extended” linear cyclic block codes.

Another constructive option is a modification of the current information calculation unit 120, in which, before calculating the current syndrome s _{m + 1, j} , the number of the j-th line is compared with the line numbers of the vector N _{m, i} . If a string match is found, then the current syndrome is not calculated. This design solution allows to increase the speed of the decoder.

Another constructive option is a modification of the current information calculation unit 120, in which the newly calculated current syndrome s _{m + 1, j is} compared with all current syndromes. If a coincidence of current syndromes is detected, then this syndrome is excluded from further processing. This design solution allows to increase the speed of the decoder.

The implementation of the described method can be hardware, software or hardware-software and can be used for both binary and q-ary codes.

Achievable technical result of the proposed method for decoding a cyclic linear block code is to increase the reliability (quality) of decoding with both “hard” and “soft” solutions, as well as increase speed and reduce hardware complexity compared to known decoding methods [1-3] .

Bibliography

1. Morelos-Zaragoza R. The art of error-correcting coding. Methods, Algorithms, Application: Per. from English V.B.Afanasyeva. - M .: Technosphere, 2005.

2. Gallager R. Information Theory and Reliable Communication: Per. from English, ed. M.S. Pinsker and B.S. Tsybakov, M., "Soviet Radio", 1974.

3. Viterbi A.D., Omura J.K. The principles of digital communication and coding: Per. from English - M.: Radio and Communications, 1982.

Claims (4)

1. The method of syndromic decoding of a cyclic linear block code, which consists in calculating a syndrome for a distorted codeword, then selecting a correction vector corresponding to it, characterized in that for the received distorted codeword, an expanded syndrome is calculated, which is represented as linear combinations of extended strings the verification matrix, and each combination uniquely corresponds to the error vector, then for each error vector the metric is calculated, which when decoding with "hard" Addressing is by its weight, and the most probable error correction as the correction vector is selected error vector, which has the minimum metric, then invert the elements in a systematic part of the codeword whose numbers correspond to the positions of non-zero elements of the correction vector. 2. The method according to claim 1, characterized in that in the case of decoding "shortened" codes with a "hard" solution, zero elements are added to the beginning of each distorted codeword, the number of which is equal to the shortening value. 3. The method of syndromic decoding of a cyclic linear block code, which consists in calculating a syndrome for a distorted codeword, then selecting a correction vector corresponding to it, characterized in that for the received distorted codeword, an expanded syndrome is calculated, which is represented as linear combinations of extended strings the verification matrix, and each combination uniquely corresponds to an error vector, then for each error vector when decoding with a “soft” solution, the mod ovannuyu metric for the most likely error correction as correction vector selected error vector, which has the maximum modified metric further inverter element in the systematic part of the codeword numbers of which correspond to the positions of non-zero elements of the correction vector. 4. The method according to claim 3, characterized in that in the case of decoding "shortened" codes with a "soft" solution, zero elements are added to the beginning of each distorted codeword, the number of which is equal to the value of the shortening with conditional probabilities equal to one.