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 R n and complex space C n . These lead to denser lattice packings than any previously known in R 36 , R 64 , R 80 , …, R 128 , …. A sequence of lattices is constructed in R n for n = 24 m ≦ 98328 (where m is an integer) for which the density Δ satisfies log 2 Δ ≈ – (1.25 …) n , and another sequence in R n for n = 2 m ( m any integer) with The 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 2 16 the best lattices found are about 2 131000 times as dense as any previously known. Minkowski proved in 1905 (see [ 20 ] and Eq. (23) below) that lattices exist with log 2 Δ > – n as n → ∞, but no infinite family of lattices with this density has yet been constructed.