Improved Lower Bounds for Kissing Numbers in Dimensions 25 through 31

Abstract

Abstract. The best previous lower bounds for kissing numbers in dimensions 25-31 were constructed using a set S with |S| = 480 of minimal vectors of the Leech Lattice, Λ 24 , such that x, y ≤ 1 for any distinct x, y ∈ S. Then, a probabilistic argument based on applying automorphisms of Λ 24 gives more disjoint sets S i of minimal vectors of Λ 24 with the same property. Cohn, Jiao, Kumar, and Torquato proved that these subsets give kissing configurations in dimensions 25-31 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 R 25 through R 31 .

High-dimensional sphere packing and the modular bootstrap

“…It is also our hope that the methods in this section (in particular, the application of the Christoffel-Darboux formula to produce positive auxiliary functions) will inspire new analytic approaches to the conformal bootstrap. The implied kissing number in high dimensions, compared with the record kissing number [21,71] and the excluded volume lower bound for the kissing number from [72].…”

“…Figure 5.2:The implied kissing number in high dimensions, compared with the record kissing number[21,71] and the excluded volume lower bound for the kissing number from[72].…”