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New Lattice Packings of Spheres

Published online by Cambridge University Press:  20 November 2018

E. S. Barnes  and
N. J. A. Sloane
E. S. Barnes
Affiliation:
The University of Adelaide, Adelaide, Australia
N. J. A. Sloane
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
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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) 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 1 , 01 February 1983 , pp. 117 - 130
Copyright
Copyright © Canadian Mathematical Society 1983

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