Reed–Muller Codes: Theory and Algorithms

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They used in many areas coding theory both electrical engineering computer science. Yet, their important properties still under investigation. This paper covers some recent developments regarding weight enumerator capacity-achieving RM codes, as well algorithmic developments. In particular, discusses connections established between thresholds Boolean functions, polarization theory, hypercontractivity, techniques approximating low codewords using lower degree polynomials (when viewed evaluation vectors r m variables). It then overviews algorithms for decoding with provable performance guarantees every block length, state-of-the-art performances practical regimes, which do not perform large length. Finally, concludes a few open problems.