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 ℝ n of density at least cn2 −n exceeding the classical Minkowski bound by a factor linear in n. This matches up to a constant the best known lower bound on the density of sphere packings due to Ball. However, our proof is very different from the earlier constructions of Minkowski, Hlawka, Rogers, and Ball. Moreover, this proof 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.