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IEEE Transactions on Information Theory

Soft-decision majority decoding of Reed-Muller codes

Ilya I. Dumer, R.E. Krichevskiy

Published Jan 1, 2000• 31 citations
Publisher

Abstract

We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.

DOI
10.1109/18.817523

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We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.
We present a new soft decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of all received symbols are recalculated into the reliabilities of the parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to the more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft decision algorithm outperforms its conventional hard decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output bit error rate /spl epsi/<1/2.
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
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.
Reed-Muller (RM) codes (m, r) of length 2/sup m/ are considered on a binary symmetric (BS) channel with high crossover error probability 1/2 -/spl epsiv/. For an arbitrarily small /spl epsiv/>0, new recursive decoding algorithms are designed that retrieve all information bits of RM codes of fixed order r with a vanishing error probability and sublinear complexity of order O(m/sup r+1/). The algorithms utilize a vanishing fraction of the received symbols for both hard- and soft-decision decoding.
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
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.