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Decoding Reed-Muller Codes Over Product Sets

Authors John Y. Kim, Swastik Kopparty

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LIPIcs.CCC.2016.11.pdf
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Keywords
  • polynomial codes
  • Reed-Muller codes
  • coding theory
  • error-correcting codes

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Abstract

We give a polynomial time algorithm to decode multivariate polynomial codes of degree d up to half their minimum distance, when the evaluation points are an arbitrary product set S^m, for every d < |S|. Previously known algorithms could achieve this only if the set S has some very special algebraic structure, or if the degree d is significantly smaller than |S|. We also give a near-linear time algorithm, which is based on tools from list-decoding, to decode these codes from nearly half their minimum distance, provided d < (1-epsilon)|S| for constant epsilon > 0.

Our result gives an m-dimensional generalization of the well known decoding algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic version of the Schwartz-Zippel lemma.

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John Y. Kim and Swastik Kopparty. Decoding Reed-Muller Codes Over Product Sets. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 11:1-11:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.CCC.2016.11

Author Details

John Y. Kim
Swastik Kopparty
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