Improving the semidefinite programming bound for the kissing number by exploiting polynomial symmetry

TL;DR

This paper enhances the upper bounds for the kissing number in dimensions 9 to 23 by exploiting polynomial symmetry in semidefinite programming bounds, improving upon previous results.

Contribution

It introduces a method to utilize polynomial symmetry in semidefinite programming to tighten bounds on the kissing number.

Findings

Improved upper bounds for dimensions 9 to 23.

Demonstrated the effectiveness of symmetry exploitation in SDP bounds.

Achieved tighter bounds than previous methods.

Abstract

The kissing number of $\mathbb{R}^n$ is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin (2010), based on the semidefinite programming bound of Bachoc and Vallentin (2008), computed the best known upper bounds for the kissing number for several values of $n \leq 23$. In this paper, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for $n = 9, \ldots, 23$.