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IEEE Transactions on Information Theory

Recursive Decoding and Its Performance for Low-Rate Reed–Muller Codes

Ilya I. Dumer

Published May 1, 2004• 98 citations
Publisher

Abstract

Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity. To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length n grows.

DOI
10.1109/tit.2004.826632

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Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length n and fixed order r. An algorithm is designed that has complexity of order nlogn and corrects most error patterns of weight up to n(1/2-/spl epsiv/) given that /spl epsiv/ exceeds n/sup -1/2r/. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity. To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length n grows.
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.
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.
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