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

RESUMO

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