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International Symposium on Turbo Codes and Iterative Information Processing

Sum-product decoding of BCH codes

Haruo Ogiwara, Kyouhei Shimamura, Toshiyuki Shohon

Published Sep 1, 2008• 2 citations
Publisher

Abstract

This paper proposes methods to improve soft-input and soft-output decoding performance of BCH codes by sum-product algorithm (SPA). A method to remove cycles of length four (RmFC) in the Tanner graph has been proposed. However, the RmFC can not realize good decoding performance for BCH codes which have more than one error correcting capability. To overcome this problem, this paper proposes two methods. One is to use a parity check matrix of the echelon canonical form as the starting check matrix of RmFC. The other is to use a parity check matrix that is concatenation (ConC) of multiple parity check matrices. For BCH(31,11,11) code, SPA with ConC realizes Eb/N03.8dB better than the original SPA, and 3.1dB better than the SPA with only RmFC at bit error rate 10−5. The decoding complexity is 0.08 times of that of Chase-2 decoding.

DOI
10.1109/turbocoding.2008.4658728

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This paper proposes methods to improve soft-input and soft-output decoding performance of BCH codes by sum-product algorithm (SPA). A method to remove cycles of length four (RmFC) in the Tanner graph has been proposed. However, the RmFC can not realize good decoding performance for BCH codes which have more than one error correcting capability. To overcome this problem, this paper proposes two methods. One is to use a parity check matrix of the echelon canonical form as the starting check matrix of RmFC. The other is to use a parity check matrix that is concatenation (ConC) of multiple parity check matrices. For BCH(31,11,11) code, SPA with ConC realizes Eb/N03.8dB better than the original SPA, and 3.1dB better than the SPA with only RmFC at bit error rate 10−5. The decoding complexity is 0.08 times of that of Chase-2 decoding.
This paper proposes methods to improve soft-input and soft-output decoding performance of BCH codes by sum-product algorithm (SPA). A method to remove cycles of length four (RmFC) in the Tanner graph has been proposed. However, the RmFC can not realize good decoding performance for BCH codes which have more than one error correcting capability. To overcome this problem, this paper proposes two methods. One is to use a parity check matrix of the echelon canonical form as the starting check matrix of RmFC. The other is to use a parity check matrix that is concatenation (ConC) of multiple parity check matrices. For BCH(31,11,11) code, SPA with ConC realizes Eb/No 3.7dB better at bit error rate 10-5 than the original SPA, and 3.1dB better than the SPA with only RmFC.
This article proposes two methods to improve the sum-product soft-in/soft-out decoding performance of convolutional codes. The first method is to transform a parity check equation in such a way as to remove cycles of length four in a Tanner graph of a convolutional code, and performs sum-product algorithm (SPA) with the transformed parity check equation. This method improves the performance of (7,5)8convolutional code (CC1). However, for (45,73)8convolutional code (CC2), the method does not effect. The second proposed method is to use a higher order parity check equation in comparison with a normal parity check equation for SPA decoding. This method improves the performance for both convolutional codes (CC1, CC2). The performance is close to that by BCJR algorithm and it is less complex than BCJR algorithm.
Binary cyclic codes achieve good error correction performance and allow the implementation of very simple encoder and decoder circuits. Among them, BCH codes represent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that soft-decision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on "Extended Parity-Check Matrices", and show that such method is not suitable for high rates or long codes. We propose a new method, based on "Spread Parity-Check Matrices", that has the same complexity, but can achieve better correction performance in many practical cases.
Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.
The authors deal with the sum–product algorithm (SPA) based on the hyperbolic tangent (tanh) rule when it is applied for decoding low-density parity-check (LDPC) codes. Motivated by the finding that, because of the large number of multiplications required by the algorithm, an overflow in the decoder may occur, two novel modifications of the tanh function (and its inverse) are proposed. By means of computer simulations, both methods are evaluated using random-based LDPC codes with binary phase shift keying (BPSK) signals transmitted over the additive white Gaussian noise (AWGN) channel. It is shown that the proposed modifications improve the bit error rate (BER) performance up to 1 dB with respect to the conventional SPA. These results have also shown that the error floor is removed at BER lower than 10−6. Furthermore, two novel approximations are presented to reduce the computational complexity of the tanh function (and its inverse), based on either a piecewise linear function or a quantisation table. It is shown that the proposed approximations can slightly improve the BER performance (up to 0.13 dB) in the former case, whereas small BER performance degradation is observed (<0.25 dB) in the latter case. In both cases, however, the decoding complexity is reduced significantly.
In the past few years, deep learning has been widely used in various fields due to its outstanding progress. One of the latest applications of deep learning is to use a neural network (NN) with trainable multiplicative weights to design decoders for error-correcting codes. High quality data are essential for deep learning to train robust NN models. In this study, two novel semi-/auto-adaptive SNR algorithms are proposed to efficiently train the neural decoders based on the Sum-Product Algorithm (SPA). For illustration, several neural SPA decoders for the Bose-Chaudhuri-Hocquenghem (BCH) code and low-density parity-check (LDPC) code have been constructed as examples. Simulation results show that, compared with the original neural decoders, the performance of these neural decoders trained by the proposed algorithms can be improved in the range of 0.2 to 0.6 dB. Moreover, the training time required for these decoders to achieve convergence can be reduced by up to 28.8% for the BCH code, and up to 35.6% for the LDPC code, without increasing decoding complexity.