A Lower Bound on the Density of Sphere Packings via Graph Theory

Abstract

Using graph-theoretic methods we give a new proof that for all sufficiently large $n$, there exist sphere packings in ${\mathbb{R}}^n$ of density at least $cn{2}^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$. This matches up to a constant the best known lower bounds on the density of sphere packings due to Rogers, Davenport-Rogers, and Ball. The suggested method makes it possible to describe the points of such a packing with complexity $\exp (n\log n)$, which is significantly lower than in the other approaches.

6 pages, 2 postscript figures