List Decoding Barnes-Wall Lattices

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 linear 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 ⊆ R^N, given a target vector r ∈ R^N and a distance parameter d, output the set of all lattice points w ∈ 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 listdecoding 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 sufuciently many processors can run in parallel time only poly-logarithmic in the lattice dimension. In particular, our results imply a polynomial-time listdecoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.