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.