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2016

Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31

Kallal, Kenz, Kan, Tomoka, Wang, Eric

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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.

Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31 · Around

Built on

Vladimir I. Levenstein, On bounds for packings in n n -dimensional Euclidian space , Soviet Math. Dokl. 20
1979
Earlier work this paper cites.
Andrew M. Odlyzko and Neil J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n n dimensions , Journal of Combinatorial Theory, Series A 26
1979
Earlier work this paper cites.

Similar

Thomas M. Thompson, From error correcting codes through sphere packings to simple groups , Mathematical Association of America, 1983
Original
1983
Cited alongside, same era.
Frank Celler, Charles R. Leedham-Green, Scott H. Murray, Alice C. Niemeyer, and Eamonn A. O’Brien, Generating random elements of a finite group , Communications in Algebra 23
1995
Cited alongside, same era.

Then

John H. Conway and Neil J. A. Sloane, Sphere packings, lattices and groups , third ed., Springer, 1999
1999
Later among the works it cites.
Henry Cohn, Yang Jiao, Abhinav Kumar, and Salvatore Torquato, Rigidity of spherical codes , Geometry & Topology 15
2011
Later among the works it cites.

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