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DOI: 10.4230/LIPIcs.CCC.2016.11

Decoding Reed-Muller Codes Over Product Sets

Authors: John Y. Kim and Swastik Kopparty

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

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)

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@InProceedings{kim_et_al:LIPIcs.CCC.2016.11,
  author =	{Kim, John Y. and Kopparty, Swastik},
  title =	{{Decoding Reed-Muller Codes Over Product Sets}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{11:1--11:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.11},
  URN =		{urn:nbn:de:0030-drops-58352},
  doi =		{10.4230/LIPIcs.CCC.2016.11},
  annote =	{Keywords: polynomial codes, Reed-Muller codes, coding theory, error-correcting codes}
}

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Decoding Reed-Muller Codes Over Product Sets

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