An indexed set of density bounds on lattice packings

Abstract

Abstract. It has long been known that the admissibility of a lattice F with respect to a symmetric convex body B is equivalent to F being a packing lattice for ½ B. This fact is the basis of the interplay between the classical theory of the arithmetic minima of positive definite quadratic forms, on the one hand, and the dense lattice packing of spheres in R '~, on the other.We give an indexed set of bounds 6z(B) > aj, where 0 < j < n/2, on the lattice packing density of B. The case j = 0 reduces to the aforementioned long-known fact, and j = 1 was proved by Elkies, Odlyzko, and Rush, and was used to obtain record high packing densities for various superballs. The new cases make possible the use of smaller primes in the construction of these dense packings.Mathematics Subject Classification (1991): 11H31.

On the lower bound for packing densities of superballs in high dimensions

“…Moreover, Elkies et al [6] improved the Minkowski-Hlawka bound exponentially for superballs and reals p > 2, and they also obtained lower bound for the packing density of more general bodies. For the lower bound constructed by error correcting codes, see Rush [20,21,22] and Liu and Xing [14].…”