Efficient Maximum-Likelihood Decoding of Reed–Muller RM(m−3,m) Codes

Abstract

Reed–Muller (RM) codes, a classical family of codes known for their elegant algebraic structure, have recently been shown to achieve capacity under maximum-likelihood (ML) decoding on the binary erasure channel and this has rekindled interest in their efficient decoding. We consider the code family RM(m−3,m) and develop a new ML decoder, for transmission over the binary symmetric channel, that exploits their large symmetry group. The new decoder has lower complexity than an earlier method introduced by Seroussi and Lempel in 1983.