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<∣S∣. 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 ∣S∣. We also
give a near-linear time randomized algorithm, which is based on tools from
list-decoding, to decode these codes from nearly half their minimum distance,
provided d0.
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.Comment: 25 pages, 0 figure