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 S-m, for every d < vertical bar S vertical bar. 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 vertical bar S vertical bar. 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) 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.