Decoding Reed-Muller Codes Using Minimum-Weight Parity Checks

TL;DR

This paper investigates exploiting the symmetry of Reed-Muller codes with minimum-weight parity checks to achieve near-ML decoding performance across various channels, improving efficiency and accuracy in short block lengths.

Contribution

It introduces a decoding approach using a parity-check matrix of minimum-weight dual codewords, enhancing performance and reducing complexity compared to traditional methods.

Findings

Near-ML performance achieved for short block lengths

Selective use of minimum-weight PCs improves decoding efficiency

Method applicable across multiple channel types

Abstract

Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML) decoding due to their highly-symmetric structure. In this paper, we explore the question of whether the code symmetry of RM codes can also be exploited to achieve near-ML performance in practice. The main idea is to apply iterative decoding to a highly-redundant parity-check (PC) matrix that contains only the minimum-weight dual codewords as rows. As examples, we consider the peeling decoder for the binary erasure channel, linear-programming and belief propagation (BP) decoding for the binary-input additive white Gaussian noise channel, and bit-flipping and BP decoding for the binary symmetric channel. For short block lengths, it is shown that near-ML performance can indeed be achieved in many cases. We also propose a method to tailor the PC matrix to the received observation by selecting only a small fraction of useful minimum-weight PCs before decoding begins. This allows one to both improve performance and significantly reduce complexity compared to using the full set of minimum-weight PCs.