Lattice (List) Decoding Near Minkowski’s Inequality

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.