Lattices with exponentially large kissing numbers

TL;DR

This paper constructs high-dimensional lattices with exponentially large kissing numbers and establishes lower bounds for the maximum lattice kissing number in n dimensions, advancing understanding of sphere packings.

Contribution

It introduces a sequence of lattices with exponentially large kissing numbers and provides new lower bounds for the maximum lattice kissing number in high dimensions.

Findings

Constructed lattices with $ au(L_{n_i})$ growing exponentially in dimension.

Established lower bounds for the maximum lattice kissing number $ au^l_{n}$ in high dimensions.

Demonstrated that $ au(L_{n_i})$ exceeds $2^{0.0338 n_i}$ asymptotically.

Abstract

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 $\log_2\tau^l_{n}> 0.0219\cdot n -o(n)$.