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 ${\mathbb{R}^{N}}$, 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 ${\mathcal{L}\subseteq \mathbb{R}^N}$, given a target vector ${r \in \mathbb{R}^N}$ and a distance parameter d, output the set of all lattice points ${w \in \mathcal{L}}$ 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 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.