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

TL;DR

This paper introduces a graph-theoretic approach to establish a new lower bound on sphere packing density in high dimensions, surpassing classical bounds and offering a more efficient description of packings.

Contribution

It provides a novel graph-theoretic proof for high-dimensional sphere packing density bounds that improves upon classical results and reduces complexity of packing descriptions.

Findings

Sphere packings exist with density at least $cn2^{-n}$ for large $n$

New proof surpasses Minkowski bound by a linear factor in $n$

Packing descriptions have complexity $ ext{exp}(n ext{log} n)$

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 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.