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IEEE International Conference on Acoustics Speech and Signal Processing

Towards Practical Near-Maximum-Likelihood Decoding of Error-Correcting Codes: An Overview

Thibaud Tonnellier, Marzieh Hashemipour, Nghia Doan, Warren J. Gross, Alexios Balatsoukas‐Stimming

Published May 13, 2021• 11 citations
Publisher

Abstract

While in the past several decades the trend to go towards increasing error-correcting code lengths was predominant to get closer to the Shannon limit, applications that require short block length are developing. Therefore, decoding techniques that can achieve near-maximum-likelihood (near-ML) are gaining momentum. This overview paper surveys recent progress in this emerging field by reviewing the GRAND algorithm, linear programming decoding, machine-learning aided decoding and the recursive projection-aggregation decoding algorithm. For each of the decoding algorithms, both algorithmic and hardware implementations are considered, and future research directions are outlined.

DOI
10.1109/icassp39728.2021.9414311

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Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial focuses on the performance evaluation of linear codes under optimal maximum-likelihood (ML) decoding. Though the ML decoding algorithm is prohibitively complex for most practical codes, their performance analysis under ML decoding allows to predict their performance without resorting to computer simulations. It also provides a benchmark for testing the sub-optimality of iterative (or other practical) decoding algorithms. This analysis also establishes the goodness of linear codes (or ensembles), determined by the gap between their achievable rates under optimal ML decoding and information theoretical limits. In Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial, upper and lower bounds on the error probability of linear codes under ML decoding are surveyed and applied to codes and ensembles of codes on graphs. For upper bounds, we discuss various bounds where focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, and also consider sphere-packing bounds with their recent improvements targeting codes of moderate block lengths. Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial is a comprehensive introduction to this important topic for students, practitioners and researchers working in communications and information theory.
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The maximum likelihood (ML) decoding is essential for error-correcting codes, but the ML decoding problem is NP-hard in general. This suggests that there is no single universal ML decoding algorithm that works efficiently for an arbitrary code. Therefore researchers developed many ML decoding algorithms with different characteristics so that they can be used complementary. This study is intended to propose an ML decoding algorithm that can be used for general non-binary linear codes. The algorithm proposed in this paper is a generalization of the adaptive and recursive ML decoding algorithm that was studied for binary codes. This paper shows that the structure and properties of linear codes that helped realizing the decoding algorithm for binary codes can be generalized to nonbinary cases, and reports the implementation results of the soft-decision ML decoding alogorithm for several non-binary linear codes.
Upper and lower bounds on the error probability of linear codes under maximum-likelihood (ML) decoding are shortly surveyed and applied to ensembles of codes on graphs. For upper bounds, focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, sphere-packing bounds, and information-theoretic bounds on the bit error probability of codes defined on graphs. A comprehensive overview is provided in a monograph by the authors which is currently in preparation.
Maximum likelihood (ML) decoding is the optimal error-correction method but is NP-hard in general, implying that there would be no polynomial-time ML decoding algorithm that capable of decoding arbitrary linear codes. Consequently, researchers have developed a variety of ML decoding algorithms with different characteristics so that they can be used in a complementary manner. Although many modern codes such as Turbo, LDPC, and Polar codes, can be efficiently decoded without ML algorithms, there remains continual demand for developing new ML decoding algorithms and evaluating ML decoding performance in various contexts. In this study, we propose an ML decoding algorithm applicable to general non-binary linear codes, formulated as a generalization of the adaptive and recursive ML decoding algorithm previously developed for binary codes. The proposed algorithm is implemented and applied to Generalized Reed-Muller codes, and its computational complexity of soft-decision ML decoding is discussed and compared with the complexity of the Viterbi decoder, in terms of the number of addition-equivalent operations, to demonstrate the efficiency of the approach. The substantial reduction in decoding complexity enables us performing practical decoding simulation of some classes of Generalized Reed-Muller codes. This paper also shows numerical evaluation results of the error-correcting capabilities of the codes, providing insights into their performance under optimal decoding conditions.