Reed-Muller Codes: Theory and Algorithms

TL;DR

This paper reviews recent theoretical and algorithmic advances in Reed-Muller codes, highlighting their properties, connections to Boolean functions, and practical decoding algorithms, while also discussing open research questions.

Contribution

It synthesizes recent developments in understanding Reed-Muller codes' properties and algorithms, connecting them to Boolean function thresholds and polarization theory.

Findings

Recent links between RM codes and Boolean function thresholds

Algorithms with performance guarantees for decoding RM codes

State-of-the-art practical decoding algorithms for RM codes

Abstract

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials. It then overviews some of the algorithms with performance guarantees, as well as some of the algorithms with state-of-the-art performances in practical regimes. Finally, the paper concludes with a few open problems.