List Decoding Barnes-Wall Lattices

TL;DR

This paper develops polynomial-time list decoding algorithms for Barnes-Wall lattices, providing tight bounds on list size and enabling efficient decoding beyond the Johnson radius.

Contribution

It introduces the first polynomial-time list decoding algorithm for Barnes-Wall lattices with tight bounds on list size, advancing lattice decoding theory.

Findings

List size is polynomial in lattice dimension for errors below a certain radius.

New list decoding algorithm runs in polynomial time and is highly parallelizable.

Achieves decoding beyond the Johnson radius, surpassing previous barriers.

Abstract

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in R^N, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L in R^N, given a target vector r in R^N and a distance parameter d, output the set of all lattice points w in L that are within distance d of r. In this work we focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes-Wall lattices. Our main contributions are twofold: 1) We give tight (up to polynomials) combinatorial bounds on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattice's minimum distance (in the Euclidean norm). 2) Building on the unique decoding algorithm of Micciancio and Nicolosi (ISIT '08), we give a list-decoding algorithm that runs in time polynomial in the lattice dimension and worst-case list size, for any error radius. Moreover, our algorithm is highly parallelizable, and with sufficiently many processors can run in parallel time only poly-logarithmic in the lattice dimension. In particular, our results imply a polynomial-time list-decoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for error-correcting codes posed by the Johnson radius.