Lattices with exponentially large kissing numbers

Abstract

We construct a sequence of lattices $\{L_{n_i}\subset {\mathbb{R}}^{n_i}\}$ for $n_i⟶\infty$, 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 ${\tau }_n^l$ in $n$ dimensions verifies ${\log }_2{\tau }_n^l>0.0219\cdot n-o(n)$.

Unfortunately, the main result of the paper should be considered as not proven. Namely, the constructions D and E applied to the codes with many light vectors give some lattices, but there is no proof that their kissing numbers will be exponential. The initial impression that those lattices retain all 0-1 short vectors from the least code $C_0$ in the tower, is definitely false

References in corpus (1)

• Curves of every genus with many points, II: Asymptotically good families

Cited by in corpus (1)

• Contact graphs of ball packings