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DOI: 10.4230/LIPIcs.CCC.2016.11
URN: urn:nbn:de:0030-drops-58352
URL: https://drops.dagstuhl.de/opus/volltexte/2016/5835/
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 | |
| Seminar: | 31st Conference on Computational Complexity (CCC 2016) | |
| Issue date: | 2016 | |
| Date of publication: | 19.05.2016 |
