The list-decoding size of reed-muller codes

In this work we study the list-decoding size of Reed-Muller codes. 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. Previous bounds of Gopalan, Klivans and Zuckerman [4] on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for all distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura [8] on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to all distances.

Weight distribution and list-decoding size of Reed-Muller codes

Abstract: We study the weight distribution and list-decoding size of Reed-Muller codes. 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 [1] 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. List-decoding has numerous theoretical and practical applications in various fields. To name a few, hardness amplification in complexity [14], constructing hard-core predicates from one way functions in cryptography [4] and learning parities with noise in learning theory [9]. Many algorithms for list-decoding such as [4, 5] as well as [14] have the crux of their analysis lying in bounding the list-decoding size. The case for Reed–Muller codes is similar, and Gopalan et. al [6] gave a list-decoding algorithm, whose complexity is determined by the list-decoding size. Gopalan et. al provided bounds on the list-decoding size of Reed–Muller codes which apply only up to the minimal distance of the code. We provide asymptotically tight bounds for the list-decoding size of Reed–Muller codes which apply for all distances