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 = 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 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.