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

For all prime powers $q\geq5$, 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