High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

摘要

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 + . . . .