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arXiv:0905.0079 · IEEE Transactions on Communications

Multiple-bases belief-propagation decoding of high-density cyclic codes

Thorsten Hehn, Johannes B. Huber, Olgica Milenković, Stefan Laendner

Published Jan 1, 2010• 49 citations
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Abstract

We introduce a new method for decoding short and moderate-length linear block codes with dense parity check matrix representations of cyclic form. This approach is termed multiple-bases belief-propagation. The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the multiple-bases belief propagation decoding performance can be shown to closely follow that of the maximum-likelihood decoder.

DOI
10.1109/tcomm.2010.01.070468

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We introduce a new method for decoding short and moderate-length linear block codes with dense parity check matrix representations of cyclic form. This approach is termed multiple-bases belief-propagation. The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the multiple-bases belief propagation decoding performance can be shown to closely follow that of the maximum-likelihood decoder.
Abstract — Multiple-bases belief-propagation is a parallel decoding setup which allows for improved decoding performance when compared to standard belief-propagation. Originally designed for decoding of high-density parity-check codes in an iterative manner, this method also shows good decoding results for well-designed low-density parity-check codes when signaling over the AWGN channel. We show the applicability of this scheme to channel codes defined in the IEEE WiMAX standard. It is challenging to find sets of well-performing parity-check matrices for these codes, all of them differing from each other. We propose an algorithm which makes use of the special structure of an underlying base matrix to accomplish this task. The results are compared to codes constructed by the progressive edge-growth algorithm and to bounds from information theory.
Short algebraic codes promise low-delay data transmission and good performance results when transmitted over the additive white Gaussian noise (AWGN) channel and decoded by maximum-likelihood (ML) soft-decision decoding. One reason for this is the large minimum distance of these codes. For belief-propagation (BP) decoding, short algebraic codes show suboptimal results due to their high-density parity-check matrices. Using a collection of parity-check matrices for a given code, Multiple-Bases Belief-Propagation (MBBP) allows for decoding of many dense linear block codes with near ML performance. One convenient way to generate these matrices is using the automorphism group of a code. We point out limits of this approach and show two novel improvements, a decoder modification and a construction algorithm for parity-check matrices, which emphasize that MBBP is a more general approach and independent of the automorphism group. We use these methods to extend the field of application for MBBP to codes of moderate length. This includes codes with parity-check matrices tailored for BP decoding, in particular Progressive Edge-Growth (PEG) codes. 1
In this paper, we propose a belief-propagation (BP)-based decoder, termed the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for quantum low-density parity-check (QLDPC) codes. The key idea is to generate \emph{structured decoding diversity} by constructing multiple redundant parity-check representations via cycle-free subtree decompositions of the Tanner graph, and running BP decoding in parallel across these representations. This extends the classical Multiple-Bases Belief-Propagation (MBBP) framework to the quantum setting while preserving the linear-time complexity and efficiency of standard BP decoding, and avoids the need for super-linear post-processing. Simulation results demonstrate that MBBP-LD improves upon existing BP-based decoders, including BP with ordered statistics decoding (BP-OSD) and belief propagation with guided decimation (BPGD) across several QLDPC codes, while requiring substantially fewer total BP iterations. For bivariate bicycle codes [[144,12,12]][[144,12,12]][[144,12,12]] and [[288,12,18]][[288,12,18]][[288,12,18]], MBBP-LD achieves up to 20%20\%20% reduction in error rate compared to BPGD and up to 30%30\%30% compared to BP-OSD in the low- and moderate-error regimes. For the larger B1 code [[882,24,18≤d≤24]][[882, 24, 18 \leq d \leq 24]][[882,24,18≤d≤24]], MBBP-LD attains comparable or improved performance relative to BPGD while maintaining BP-like decoding latency under parallel implementation.
Belief propagation (BP) decoding has been recognized for its capacity-approaching performance and high throughput when decoding long low-density parity-check (LDPC) codes. However, the application of BP decoding for short codes is hindered by dense parity-check matrices (PCMs) and prevalent short cycles in the Tanner graph. In this paper, we introduce a general method to extract an optimized sparse PCM for short binary block codes, which removes length-four cycles and enhances the connectivity of short cycles to enable BP decoding with improved performance. Notably, for short binary codes with lengths up to 64, our BP decoding performance approaches the maximum likelihood bound and surpasses the best-reported BP results with reduced computational complexity. Compared with other universal decoding algorithms, BP decoding using our extracted sparse PCMs is competitive in terms of both error-rate performance and computational complexity. These promising results suggest that our method to improve BP decoding for short codes is a step toward a practical universal BP decoder for next-generation communication systems.
A novel soft-decoding method for algebraic block codes is presented. The algorithm is designed for soft-decision decoding and is based on belief-propagation (BP) decoding using multiple bases of the dual code. Compared to other approaches for high-performance BP decoding, this method is conceptually simple and does not change at each stage of the decoding process. With its multiple BP decoders the proposed scheme achieves the performance of a standard BP algorithm with a significantly lower number of iterations per decoder realization. By this means the data delay introduced by decoding is reduced. Moreover, a significant improvement in decoding performance is achieved while keeping the data delay small. It is shown that for selected codes the proposed scheme approaches near maximum likelihood (ML) performance for very small data processing delays.
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.