List-Decoding Barnes–Wall Lattices

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 linear structure of linear codes and point lattices in $${\mathbb{R}^{N}}$$RN, 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 $${\mathcal{L}\subseteq \mathbb{R}^N}$$L⊆RN, given a target vector $${r \in \mathbb{R}^N}$$rźRN and a distance parameter d, output the set of all lattice points $${w \in \mathcal{L}}$$wźL 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 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. We use our combinatorial bounds to generalize the unique-decoding algorithm of Micciancio and Nicolosi (IEEE International Symposium on Information Theory 2008) to work beyond the unique-decoding radius and still run in polynomial time up to the list-decoding radius. Just like the Micciancio---Nicolosi algorithm, our algorithm is highly parallelizable, and with sufficiently many processors, it can run in parallel time only poly-logarithmic in the lattice dimension.