Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31

Abstract

The best previous lower bounds for kissing numbers in dimensions 25 through 31 were constructed using a set $S$ with $|S| = 480$ of minimal vectors of the Leech Lattice, $\Lambda_{24}$, such that $\langle x, y \rangle \leq 1$ for any distinct $x, y \in S$. Then, a probabilistic argument based on applying automorphisms of $\Lambda_{24}$ gives more disjoint sets $S_i$ of minimal vectors of $\Lambda_{24}$ with the same property. Cohn, Jiao, Kumar, and Torquato proved that these subsets give kissing configurations in dimensions 25 through 31 of given size linear in the sizes of the subsets. We achieve $|S| = 488$ by applying simulated annealing. We also improve the aforementioned probabilistic argument in the general case. Finally, we greedily construct even larger $S_i$'s given our $S$ of size $488$, giving increased lower bounds on kissing numbers in $\mathbb{R}^{25}$ through $\mathbb{R}^{31}$.