Lattices with exponentially large kissing numbers

Abstract

We construct a sequence of lattices $\left\{L_{n_i}\subset {ℝ}^{n_i}\right\}$ for $n_i\to \infty$ with exponentially large kissing numbers, namely, ${log}_2\tau \left(L_{n_i}\right)>0.0338\cdot n_i-o\left(n_i\right)$. We also show that the maximum lattice kissing number ${\tau }_n^l$ in $n$ dimensions satisfies ${log}_2{\tau }_n^l>0.0219\cdot n-o\left(n\right)$ for any $n$.