Reed-Muller Codes: Theory and Algorithms

Emmanuel Abbe, Amir Shpilka, Min Ye*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (when codewords are viewed as evaluation vectors of degree r polynomials in m variables). It then overviews some of the algorithms for decoding RM codes. It covers both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length. Finally, the paper concludes with a few open problems.

Original languageEnglish
Article number9123985
Pages (from-to)3251-3277
Number of pages27
JournalIEEE Transactions on Information Theory
Issue number6
StatePublished - Jun 2021


  • Reed-Muller codes
  • Shannon capacity
  • decoding algorithms
  • polarization
  • weight enumerator


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