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 a weight of up to the given parameter. Additionally, 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. In this work, we make a new connection between computer science techniques used for studying low-degree polynomials and these coding theory questions. Using this connection we progress significantly towards resolving both the weight distribution and the list-decoding problems. 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 and seemingly resistent to the common coding theory tools. The best results to date are by Azumi, Kasami and Tokura which provide bounds on the weight distribution that apply only up to 2.5 times the minimal distance of the code. We provide asymptotically tight bounds for the weight distribution of the Reed-Muller code that apply to all distances