Weight Distribution and List-Decoding Size of Reed–Muller Codes

The weight distribution and list-decoding size of Reed–Muller codes are studied in this work. Given a weight parameter, we are interested in bounding the number of Reed–Muller codewords with weight up to the given parameter; and given a received word and a distance parameter, we are interested in bounding the size of the list of Reed–Muller codewords that are within that distance from the received word. Obtaining tight bounds for the weight distribution of Reed–Muller codes has been a long standing open problem in coding theory, dating back to 1976. In this work, we make a new connection between computer science techniques used to study low-degree polynomials and these coding theory questions. This allows us to resolve the weight distribution and list-decoding size of Reed–Muller codes for all distances. Previous results could only handle bounded distances: Azumi, Kasami, and Tokura gave bounds on the weight distribution which hold up to 2.5 times the minimal distance of the code; and Gopalan, Klivans, and Zuckerman gave bounds on the list-decoding size which hold up to the Johnson bound.