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Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML)decoding due to their highly-symmetric structure. In this paper, we explore thequestion of whether the code symmetry of RM codes can also be exploited toachieve near-ML performance in practice. The main idea is to apply iterativedecoding to a highly-redundant parity-check (PC) matrix that contains only theminimum-weight dual codewords as rows. As examples, we consider the peelingdecoder for the binary erasure channel, linear-programming and beliefpropagation (BP) decoding for the binary-input additive white Gaussian noisechannel, and bit-flipping and BP decoding for the binary symmetric channel. Forshort block lengths, it is shown that near-ML performance can indeed beachieved in many cases. We also propose a method to tailor the PC matrix to thereceived observation by selecting only a small fraction of usefulminimum-weight PCs before decoding begins. This allows one to both improveperformance and significantly reduce complexity compared to using the full setof minimum-weight PCs.

Decoding Reed-Muller Codes Using Minimum-Weight Parity Checks