Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31

TL;DR

This paper improves lower bounds for kissing numbers in dimensions 25 to 31 by increasing the size of specific vector sets using simulated annealing and enhanced probabilistic methods, leading to larger configurations.

Contribution

It introduces a larger set of minimal vectors of the Leech lattice and refines probabilistic techniques to establish improved lower bounds for kissing numbers in specified dimensions.

Findings

Achieved a set size of 488 vectors using simulated annealing.

Constructed larger kissing configurations in dimensions 25-31.

Provided improved lower bounds on kissing numbers in these dimensions.

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 $\mathbb{R}^{31}$.