New Lattice Packings of Spheres

Abstract

1. Introduction. In this paper we give several general constructions for lattice packings of spheres in real n-dimensional space Rn and complex space Cn. These lead to denser lattice packings than any previously known in R36, R64, R80, …, R128, …. A sequence of lattices is constructed in Rn for n = 24m ≦ 98328 (where m is an integer) for which the density Δ satisfies log2 Δ ≈ – (1.25 …)n, and another sequence in Rn for n = 2m (m any integer) withThe latter appear to be the densest lattices known in very high dimensional space. (See, however, the Remark at the end of this paper.) In dimensions around 216 the best lattices found are about 2131000 times as dense as any previously known.Minkowski proved in 1905 (see [20] and Eq. (23) below) that lattices exist with log2 Δ > –n as n → ∞, but no infinite family of lattices with this density has yet been constructed.