Lattices with exponentially large kissing numbers

We construct a sequence of lattices $\{L_{n_i}\subset \mathbb R^{n_i}\}$ for $n_i\longrightarrow\infty$, with exponentially large kissing numbers, namely, $\log_2\tau(L_{n_i})> 0.0338\cdot n_i -o(n_i)$. We also show that the maximum lattice kissing number $\tau^l_{n}$ in $n$ dimensions verifies $\log_2\tau^l_{n}> 0.0219\cdot n -o(n)$.