Lattice (List) Decoding Near Minkowski's Inequality

Abstract

Minkowski proved that any n-dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most √ n; in fact, there are 2 Ω(n) such lattice vectors. Lattices whose minimum distances come close to Minkowski's bound provide excellent sphere packings and error-correcting codes in 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( √ log n) factor of Minkowski's bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching 1/ √ 2 times 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.