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IEEE Transactions on Information Theory

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Ilya I. Dumer

Published Mar 1, 2006• 59 citations
Publisher

Abstract

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

DOI
10.1109/tit.2005.864425

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IEEE Information Theory Workshop (ITW)•Mar 1, 2004•2 citations
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
For Reed-Muller (RM) codes of length n and distance d, a recursive decoding algorithm is designed that has complexity of order nlogn for any fixed rate R, and corrects most error patterns of weight up to (dlnd)/2 on the binary symmetric channels (BSC) and up to (2dlnd)/pi on the AWGN channels. For long RM codes of fixed order r, a vanishing decoding error probability and polylogarithmic decoding complexity of order (logn)r+1are obtained on the BSC given any transition probability p that is bounded away from 1/2
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.
Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM (1, m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences wj|yj- cj| taken over all n positions. We obtain a tight upper bound LTon the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O (n ln2LT). In particular, all possible error weights wjequal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM (1, m) and requires O (n ln2n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4-1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed (0, ½), our algorithm outputs the complete list of codewords within the decoding radius n(½-) with linear complexity of order n ln2.
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.
Let a binary Reed-Muller code RM(s;m) of length n be used on a memoryless channel with an input alphabet ±1 and a real-valued output ℝ. Given a received vector y in ℝn; we define its generalized distance T to any codeword c as the sum Σ|yj|taken over all positions j, in which vectors y, c have opposite signs. We then consider the list ℒTof codewords located within distance T from the received vector y and estimate the size LTof this list using the generalized Johnson bound. For any RM code RM(s,m) of fixed order s, the algorithm is proposed that performs list decoding beyond the error-correcting radius with linear complexity in length n and retrieves the code list ℒTwith complexity of order nsLTfor any decoding radius T within the generalized Johnson bound.
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.