Theory of Computing
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Title : Decoding Reed–Muller Codes over Product Sets
Authors : John Y. Kim and Swastik Kopparty
Volume : 13
Number : 21
Pages : 1-38
URL : https://theoryofcomputing.org/articles/v013a021
Abstract
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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 $S$ had some very special algebraic structure, or if the degree $d$
was significantly smaller than $|S|$. We also give a near-linear-time
randomized algorithm, 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.
A conference version of this paper appeared in the
Proceedings of the 31st Conference on Computational
Complexity, 2016.