High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

Abstract

The kissing number in n-dimensional Euclidean space is the maximal number of nonoverlapping unit spheres that 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 sphere packing with average theta series 1 + 7680_q ³ + 4320_q ⁴ + 276480_q ⁵ + 61440_q ⁶ + . . . .

Keywords:

• 11F11, 52C17, 90C10
• kissing number
• semidefinite programming
• average theta series
• extremal modular form