High accuracy semidefinite programming bounds for kissing numbers

TL;DR

This paper computes high-precision upper bounds for the kissing number in dimensions up to 24 using semidefinite programming, confirming a conjecture about 16-dimensional sphere packings.

Contribution

It provides highly accurate semidefinite programming bounds for kissing numbers in multiple dimensions, advancing the understanding of sphere packings.

Findings

Bound for n=16 supports a conjecture about 16-dimensional sphere packings.

High-accuracy bounds for n <= 24 are computed.

Results have implications for the structure of periodic point sets.

Abstract

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ...