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Information Theory and Applications Workshop (ITA)

A Novel Coding Scheme for Encoding and Iterative Soft-Decision Decoding of Binary BCH Codes of Prime Lengths

Shu Lin, Khaled Abdel-Ghaffar, Juane Li, Keke Liu

Published Feb 1, 2018• 0 citations
Publisher

Abstract

A novel coding scheme is presented for encoding and iterative soft-decision decoding of binary BCH codes of prime lengths. The encoding of such a BCH code is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword of a nonbinary low-density parity-check (LDPC) code which has a binary parity-check matrix for transmission. Using this matrix, a binary iterative soft-decision decoding algorithm based on belief-propagation is applied to jointly decode a collection of codewords of the BCH code. The joint-decoding allows information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding a BCH code of prime length, the proposed decoding scheme not only requires low decoding complexity, but also yields superior performance. The proposed decoding scheme can achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords.

DOI
10.1109/ita.2018.8503231

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In this paper, an iterative soft-decision hybrid decoding algorithm for non-binary low-density parity-check (LDPC) codes with short codeword lengths is proposed. The rationale of the approach is to combine the classical belief propagation (BP) iterative LDPC decoding algorithm with the most reliable basis (MRB) decoding algorithm. This allows to achieve significant performance improvements, with a complexity that, for medium/low error rates, is only slightly higher than that of the BP algorithm alone. The performance improvement with respect to pure BP decoding is up to 0.7 dB at codeword error rate (CER) ≈ 10-5. Notably, for a fixed MRB order, hybrid decoding achieves a gain up to 0.5 dB at CER ≈ 10-5with respect to BP decoding and MRB decoding used alone.
In this paper, an iterative joint decoding algorithm for low density parity check codes with BCH codes based on Chase algorithm is proposed. After a certain number of iterations, the soft output values delivered by belief propagation algorithm of LDPC codes are used as reliability values to perform Chase decoding of BCH codes. And BCH decoder also will return soft information to LDPC decoder as input. This approach presents an option scheme for LDPC with medium length because of the relevant presence of short loops. It allows to bridge the gap between error performance and maximum likelihood decoding. Simulation result for the scheme is given.
In this paper, an iterative soft-decision (SD) decoding algorithm for cyclic codes based on extended parity-check equations is developed. The algorithm does not necessarily utilize the algebraic properties of the code, but operates on transforming the systematic parity-check matrix using the soft reliability information matrix obtained from the received vector. Results show a significant performance gain when compared with the hard decision Berlekamp-Massey(B-M) and belief propagation (BP) algorithms, but present a similar bit error rate (BER) performance when compared to the adaptive belief propagation (ABP) algorithm. An important feature of the decoder is that it functions within a practical decoding time complexity, and can be generally implemented for the class of linear block codes.
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.
In this paper, reliability based decoding is combined with belief propagation (BP) decoding for low-density parity check (LDPC) codes. At each iteration, the soft output values delivered by the BP algorithm are used as reliability values to perform reduced complexity soft decision decoding of the code considered. This approach allows to bridge the error performance gap between belief propagation decoding which remains suboptimum, and maximum likelihood decoding which is too complex to be implemented for the codes considered. Trade-offs between decoding complexity and error performance are also investigated. In particular, a stopping criterion which reduces the average number of iterations at the expense of very little performance degradation is proposed for this combined decoding approach. Simulation results for several Gallager (1963, 1968) LDPC codes and different set cyclic codes of hundreds of information bits are given and elaborated.
We develop a belief-propagation (BP) decoder for the joint decoding of multiple codewords which belong to the same non-binary LDPC code. Decoding is based on soft information in form of joint channel-posterior probabilities of all codeword symbols. We extend the BP algorithm for q-ary LDPC codes such that the FFT-based check node processing is preserved and the complexity remains manageable. This joint decoding is useful in settings in which multiple codewords are transmitted in a non-orthogonal way over the same channel, including multiple-access with packet collisions, physical-layer network coding and multi-resolution broadcasting. We show in an example that joint decoding can be far superior to separate decoding.