License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2016.11
URN: urn:nbn:de:0030-drops-58352
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5835/
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Kim, John Y. ; Kopparty, Swastik

Decoding Reed-Muller Codes Over Product Sets

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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.

BibTeX - Entry

@InProceedings{kim_et_al:LIPIcs:2016:5835,
  author =	{John Y. Kim and Swastik Kopparty},
  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 =	{Ran Raz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5835},
  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}
}

Keywords: polynomial codes, Reed-Muller codes, coding theory, error-correcting codes
Collection: 31st Conference on Computational Complexity (CCC 2016)
Issue Date: 2016
Date of publication: 19.05.2016

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