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arXiv:1703.05303

Recursive decoding of Reed-Muller codes

Ilya I. Dumer

Published Mar 14, 2017• 4 citations
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Abstract

New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ of length $2^{m}$ and distance $2^{m-r}$. We use Plotkin $(u,u+v)$ construction and decompose code $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ onto subblocks $u\in\left\{\genfrac{}{}{0pt}{}{m-1}{r}\right\} $ and $v\in\left\{\genfrac{}{}{0pt}{}{m-1}{r-1}\right\} .$ In decoding, we first try to find a subblock $v$ from the better protected code and then proceed with the block $u$. 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 $n\log n.$ 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 $d\ln d/2.$

DOI
10.48550/arxiv.1703.05303

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Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length(≤256)(\leq 256)(≤256)RM codes. Recursive decoding employs the Plotkin construction to split the original code into two shorter RM codes with different rates. In contrast to the standard approach which decodes the lower-rate code first, the method in this paper decodes the higher-rate code first. This modification enables an efficient permutation-based decoding technique, with permutations being selected on the fly from the automorphism group of the code using soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms, enhanced by a permutation selection technique, is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for short RM codes and for RM codes of length2m2^{m}2mand orderm−3m-3m−3with reasonable complexity.
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes
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
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes.
Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length nnn and fixed order r.r.r. An algorithm is designed that has complexity of order nlog⁡nn\log nnlogn and corrects most error patterns of weight up to n(1/2−ε)n(1/2-\varepsilon)n(1/2−ε) given that ε\varepsilonε exceeds n−1/2r.n^{-1/2^{r}}.n−1/2r. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.