Sphere packing bounds via spherical codes

TL;DR

This paper improves upper bounds on sphere packing densities in Euclidean and hyperbolic spaces using geometric and linear programming methods, extending classical bounds and establishing new theoretical results.

Contribution

It refines the Kabatiansky-Levenshtein bound with a simple geometric argument and extends bounds to hyperbolic space, also comparing linear programming bounds in both geometries.

Findings

Improved sphere packing bounds in high-dimensional Euclidean space.

Extended packing bounds to hyperbolic space with exponential improvements.

Showed Cohn-Elkies bounds are at least as strong as Kabatiansky-Levenshtein bounds.

Abstract

The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn-Elkies linear programming bound is always at least as strong as the Kabatiansky-Levenshtein bound; this result is analogous to Rodemich's theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich's theorem there as well.