Lattice (List) Decoding Near Minkowski's Inequality

TL;DR

This paper presents a polynomial-time list decoding algorithm for a family of lattices close to Minkowski's bound, using Reed-Solomon code decoding techniques to approach optimal error correction in Euclidean space.

Contribution

It introduces a novel polynomial-time list decoding method for Barnes-Sloane lattices, leveraging Reed-Solomon decoding under Euclidean error measurement.

Findings

Decodes lattices to nearly half their minimum distance.

Uses Reed-Solomon decoding techniques in Euclidean norm.

Provides efficient decoding close to Minkowski's bound.

Abstract

Minkowski proved that any $n$-dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most $\sqrt{n}$; in fact, there are $2^{\Omega(n)}$ such lattice vectors. Lattices whose minimum distances come close to Minkowski's bound provide excellent sphere packings and error-correcting codes in $\mathbb{R}^{n}$. The focus of this work is a certain family of efficiently constructible $n$-dimensional lattices due to Barnes and Sloane, whose minimum distances are within an $O(\sqrt{\log n})$ factor of Minkowski's bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching $1/\sqrt{2}$ of the minimum distance. The main technique is to decode Reed-Solomon codes under error measured in the Euclidean norm, using the Koetter-Vardy "soft decision" variant of the Guruswami-Sudan list-decoding algorithm.