List Decoding Tensor Products and Interleaved Codes

TL;DR

This paper introduces efficient algorithms and new bounds for list decoding tensor products and interleaved codes, showing that their decoding radius remains unchanged, with applications to multivariate polynomials and linear transformations.

Contribution

It provides the first efficient list decoders and combinatorial bounds for tensor and interleaved codes, demonstrating invariance of decoding radius under these operations.

Findings

List decoding radius remains unchanged under tensor products.

List decoding radius remains unchanged under m-wise interleaving.

New bounds on list size using generalized Hamming weights and weight distribution analysis.

Abstract

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for {\em 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. We show that for {\em 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 \cite{DGKS}, who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). Using the notion of generalized Hamming weights, we give better list size bounds for {\em 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 list size bounds that are tight over small fields.