A lattice family with kissing numbers $\tau ({\mathcal{L}}_n)\ge e^{2\sqrt{n}}$

Abstract

For all prime powers $q\ge 5$, we construct lattices ${\mathcal{L}}_q\subseteq {\mathbb{Z}}^q$ with kissing numbers \[ τ(\mathcal{L}_q)\geq \left(\frac{1}{2πe^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\frac{1}{2}{n}^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

arXiv commentv1 contained a construction with $\tau ({\mathcal{L}}_n)\ge e^{\sqrt{n}}$; v2 improves the constant in the exponent by a factor $2$