Dual linear programming bounds for sphere packing via modular forms

TL;DR

This paper introduces a novel method using modular forms to improve the dual linear programming bounds for sphere packing, revealing significant gaps in certain dimensions compared to known optimal packings.

Contribution

It develops a systematic approach to derive new restrictions on linear programming bounds for sphere packing using modular forms, demonstrating separations in specific dimensions.

Findings

Linear programming bounds are not tight in dimensions 12, 16, 20, 28, and 32.

The method produces feasible points in the dual linear program via modular forms.

The approach provides a systematic way to prove bounds separations in sphere packing.

Abstract

We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.