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IEEE International Symposium on Information Theory (ISIT)

Multiple-Bases Belief-Propagation for Decoding of Short Block Codes

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

Published Jun 1, 2007• 48 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2007.4557244

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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.
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
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.
Reed–Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a belief propagation (BP) decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) using polar BP as constituent decoders. We provide extensive errorrate performance simulations and compare our results to existing decoding schemes. We report a near-ML performance for the RM(3,7)-code (e.g., 0.05 dB away from the ML bound at BLER of 10−4) at a competitive computational cost. To the best of our knowledge, our proposed decoder achieves the best performance of all iterative RM decoders presented thus far.
Iterative soft decision decoding of linear block codes is a practical necessity when working with even modest block lengths. A number of algorithms have been proposed in the literature which use the permutation group of a code and the belief propagation (BP) algorithm for decoding. A novel soft-input, soft-output algorithm is presented that can be used for efficiently decoding of linear cyclic codes. Utilising the automorphism property of cyclic codes the permutation is incorporated into the belief propagation algorithm resulting in faster convergence and better error correcting performance. Performance of the new approach is analysed using a (63,45) BCH code and a (72,36) quadratic residue code. Simulation results show significant reduction in the average number of required decoding iterations and some improvement in error correcting performance over published algorithms.
We propose a belief propagation list (BPL) decoder with comparable performance to the successive cancellation list (SCL) decoder of polar codes, which already achieves the maximum likelihood (ML) bound of polar codes for sufficiently large list size L. The proposed decoder is composed of multiple parallel independent belief propagation (BP) decoders based on differently permuted polar code factor graphs. A list of possible transmitted codewords is generated and the one closest to the received vector, in terms of Euclidean distance, is picked. To the best of our knowledge, the proposed BPL decoder provides the best performance of plain polar codes under iterative decoding known so far. The proposed algorithm does not require any changes in the polar code structure itself, rendering the BPL into an alternative to the SCL decoder, equipped with a soft output capability enabling, e.g., iterative detection and decoding to further improve performance. Further benefits are the lower decoding latency than the SCL decoder and the possibility of high throughput implementations. Additionally, we show that a different selection strategy of frozen bit positions can further enhance the error-rate performance of the proposed decoder.
The problem of finding good error correcting codes for short block lenghts and its corresponding decoders is an open research topic. A frequently applied soft decoder is the Belief Propagation (BP) decoder, however with degraded performance in case of short loops in the Tanner graph. This is especially problematic for short length codes as loops of small length are more likely to occur. In this paper, we propose the Machine Learning Scaled Belief Propagation (MLS-BP) to mitigate the performance loss of BP decoding for short length codes by introducing a learned scaling factor for the receive signals. The key point of this approach is the fact that the implementation of the BP decoder is not changed and the simple scaling leads to performance results comparable to other proposed BP improvements.