List decoding tensor products and interleaved codes

Abstract

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. (1) We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. (2) We show that for every code, its list decoding radius remains unchanged under m-wise interleaving for an integer m. This generalizes a recent result of Dinur.et.al, who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). (3)Using the notion of generalized Hamming weights, we give better list size bounds for both tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of bounding the list size to bounding the number of close-by low-rank codewords. For decoding linear transformations, using rank-reduction together with other ideas, we obtain tight list size bounds for small fields.Our results give better bounds on the list decoding radius than what is obtained from the Johnson bound, and yield rather general families of codes decodable beyond the Johnson bound.