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.
Bibliographical noteFunding Information:
Manuscript received April 13, 2010; revised September 15, 2011; accepted September 20, 2011. Date of publication January 31, 2012; date of current version April 17, 2012. This work was supported in part by the Alon Fellowship and by the National Science Foundation under Grant DMS-0835373. The work of T. Kaufman was supported in part by NSF Awards CCF-0514167 and NSF-0729011. The work of S. Lovett was supported in part by the Israel Science Foundation (Grant 1300/05). This work was performed in part when the S. Lovett was an intern at Microsoft Research.
- List decoding
- Reed-Muller codes
- weight distributions