Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry

ABSTRACT

The kissing number of  is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10], based on the semidefinite programming bound of Bachoc and Vallentin [Bachoc and Vallentin 08], computed the best known upper bounds for the kissing number for several values of n ⩽ 23. In this article, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for n = 9, …, 23.

KEYWORDS:

• kissing number
• semidenite programming
• invariant semidefinite programs
• symmetric polynomials

2000 AMS Subject Classification:

• 52C17
• 90C22

Funding

The first author was supported by the São Paulo State Research Foundation (FAPESP) under grants 2015/05648-4 and 2014/16058-0. The second author was partially supported by FAPESP grant 2013/03447-6.