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

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 $e^{\mathrm{\Theta }((\log n{)}^2)}$ of the Barnes-Wall lattices, which provided the previously best known asymptotic lower bound. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.