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

For prime dimensions $n \ge 5$, we define integer lattices $\mathcal{L}_n$ with kissing constant \begin{align} τ(\mathcal{L}_n) \ge \left(\frac{1}{\sqrt{2 πe}} + o(1)\right) n^{3/4} e^{\sqrt n}. \notag \end{align} This improves on the growth rate $e^{Θ((\log n)^2)}$ of the Barnes-Wall lattices, whic