New lower bounds on kissing numbers and spherical codes in high dimensions

Abstract

Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in ${\mathbb{R}}^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject of a famous discussion between Gregory and Newton in 1694. We prove that, as the dimension $d$ goes to infinity, $$K(d)\ge (1+o(1))\frac{\sqrt{3\pi }}{4\sqrt{2}}\log \frac{3}{2}\cdot d^{3/2}\cdot (\frac{2}{\sqrt{3}}{)}^d,$$ thus improving the previously best known bound of Jenssen, Joos and Perkins by a factor of $\log (3/2)/\log (9/8)+o(1)=\mathrm{3.442...}$. Our proof is based on the novel approach from Jenssen, Joos and Perkins that uses the hard core sphere model of an appropriate fugacity. Similar constant-factor improvements in lower bounds are also obtained for general spherical codes, as well as for the expected density of random sphere packings in the Euclidean space ${\mathbb{R}}^d$.

20 pages