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IEEE International Symposium on Information Theory (ISIT)

Generalised minimum distance decoding of Reed-Muller codes and Barnes-Wall lattices

Chunxiang Wang, Ba-Zhong Shen, Kenneth K. Tzeng

Published Nov 19, 2002• 4 citations
Publisher

Abstract

Low complexity soft decision decoding algorithms for Reed-Muller codes and Barnes-Wall lattices are presented. These algorithms are constructed based on the usage of generalised minimum distance (GMD) decoding recursively. Evaluation of the algorithms on AWGN channel through computer simulation indicates a slight degradation in performance, compared to maximum likelihood decoding, but with considerable reduction in complexity.

DOI
10.1109/isit.1995.531535

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Low complexity soft decision decoding algorithms for Reed-Muller codes and Barnes-Wall lattices are presented. These algorithms are constructed based on the usage of generalised minimum distance (GMD) decoding recursively. Evaluation of the algorithms on AWGN channel through computer simulation indicates a slight degradation in performance, compared to maximum likelihood decoding, but with considerable reduction in complexity.
Constructs Reed-Muller codes by generalized multiple concatenation of binary block codes of length 2. As a consequence of this construction, a new decoding procedure is derived that uses soft-decision information. The algorithm is designed for low decoding complexity and is applicable to all Reed-Muller codes. It gives better decoding performance than soft-decision bounded-distance decoding. Its decoding complexity is much lower than that of maximum-likelihood trellis decoding of Reed-Muller codes, especially for long codes.>
New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ of length 2m2^{m}2m and distance 2m−r2^{m-r}2m−r. We use Plotkin (u,u+v)(u,u+v)(u,u+v) construction and decompose code $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ onto subblocks $u\in\left\{\genfrac{}{}{0pt}{}{m-1}{r}\right\} $ and v∈{m−1r−1}.v\in\left\{\genfrac{}{}{0pt}{}{m-1}{r-1}\right\} .v∈{r−1m−1​}. In decoding, we first try to find a subblock vvv from the better protected code and then proceed with the block uuu. The likelihoods of the received symbols are recalculated in a way similar to belief propagation. Thus, decoding is relegated to the two constituent codes. We repeat this recursion and execute decoding only at the end nodes $\left\{\genfrac{}{}{0pt}{}{j}{1}\right\} $ and $\left\{\genfrac{}{}{0pt}{}{j}{j-1}\right\} $. The overall complexity has low order of nlog⁡n.n\log n.nlogn. It is shown that this decoding substantially outperforms other algorithms of polynomial complexity known for RM codes. In particular, for medium and high code rates, the algorithm corrects most error patterns of weight dln⁡d/2.d\ln d/2.dlnd/2.
Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart
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.