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IEEE Transactions on Information Theory

Error Exponents for Recursive Decoding of Reed–Muller Codes on a Binary-Symmetric Channel

M. V. Burnashev, Ilya I. Dumer

Published Oct 30, 2006• 13 citations
Publisher

Abstract

Error exponents are studied for recursive decoding of Reed-Muller (RM) codes and their subcodes used on a binary-symmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs plusmn1 are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of 256. It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code

DOI
10.1109/tit.2006.883557

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Error exponents are studied for recursive decoding of Reed-Muller (RM) codes and their subcodes used on a binary-symmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs plusmn1 are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of 256. It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code
Recursive decoding is studied for Reed-Muller (RM) codes used on a binary symmetric channel. Decoding is performed beyond the bounded distance radius d/2 and corrects most error patterns of weight up to (dlnd)/2. In our analysis, decoding is decomposed into consecutive steps, with one information bit derived in each step. Then the error probability of each step is defined by the recursive recalculations of the Bernoulli random variables. We derive the exponential moments of the recalculated random variables. As a result, tight exponential bounds on the output error probability are obtained for the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results
Error exponents are studied for recursive and majority decoding of general Reed-Muller (RM) codesRM(r,m) used on the additive white Gaussian noise (AWGN) channels. Both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols and determine different information symbols using soft-decision majority voting. For any codeRM(r,m), we estimate the probabilities of the information symbols obtained in these recalculations and derive the analytical upper bounds for the block error rates of the recursive and majority decoding. In the case of a low noise, we also obtain the lower bounds and show that the upper bounds are tight. For a higher noise, these bounds closely approach our simulation results.
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.