ICS 2010

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Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 5-7, 2010. Proceedings, 422-433, 978-7-302-21752-7 Tsinghua University Press

Weight Distribution and List-Decoding Size of Reed-Muller Codes Authors :Tali Kaufman, Shachar Lovett and Ely Porat [Download PDF]

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 [15], constructing hard-core predicates from one way functions in cryptography [4] and learning parities with noise in learning theory [10]. Many algorithms for list-decoding such as [4, 6] as well as [15] 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 [7] 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.

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