Decoding Reed-Muller Codes Over Product Sets

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 Sm, for every d < IS1. Previously known algorithms can achieve this only if the set S has some very special algebraic structure, or if the degree d is significantly smaller than vertical bar S vertical bar. 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)vertical bar S vertical bar 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