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

TL;DR

This paper establishes new lower bounds on the kissing number in high dimensions, significantly improving previous estimates by employing a novel approach based on the hard core sphere model.

Contribution

It introduces a new method to derive tighter lower bounds on kissing numbers and spherical codes in high dimensions, advancing the understanding of sphere packing limits.

Findings

Derived a lower bound for $K(d)$ with a factor of 3.442 improvement.

Extended the approach to general spherical codes.

Provided bounds on the density of random sphere packings.

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, $$ K(d)\ge (1+o(1)){\frac{\sqrt{3\pi}}{4\sqrt2}}\,\log\frac{3}{2}\cdot d^{3/2}\cdot \Big(\frac{2}{\sqrt{3}}\Big)^{d}, $$ thus improving the previously best known bound of Jenssen, Joos and Perkins by a factor of $\log(3/2)/\log(9/8)+o(1)=3.442...$. 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 $\mathbb R^d$.