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Improved iterative decoding of LDPC codes from the IEEE WiMAX standard

Thorsten Hehn, Johannes B. Huber, Stefan Laendner

Published Jan 1, 2010• 9 citations
OpenAlex

Abstract

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.

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arXiv:1605.04661•IEEE Information Theory Workshop (ITW)•Nov 1, 2014•6 citations
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.
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.
At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.
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.
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
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.