Dense lattices in low dimensions

TL;DR

This paper constructs new 16-dimensional lattices with densities matching the Barnes-Wall lattice, challenging the conjecture that it is the unique densest lattice in that dimension, and also introduces new lattices in lower dimensions.

Contribution

It introduces two new 16-dimensional lattices with the same density but different kissing numbers, and derives new lattices in 14 and 15 dimensions, disproving previous uniqueness conjectures.

Findings

New 16-dimensional lattices with density 1/16 and kissing numbers 4224 and 4176.

Existence of multiple densest lattices in dimensions 14 and 15.

Counterexamples to the uniqueness of Barnes-Wall lattice in dimension 16.

Abstract

The Barnes-Wall lattice ${\bf \Lambda}_{16}$ with the center density ${\{1}{16}}$ and the kissing number 4320 was found in 1959 and is the only known densest sphere packing in the dimension 16. J. H. Conway and N.J.A. Sloane conjectured that ${\bf \Lambda}_{16}$ is the densest 16 dimensional lattice. Sometimes it is conjectured that the Barnes-Wall lattice ${\bf \Lambda}_{16}$ is the only densest lattice and the optimal sphere packing in ${\bf R}^{16}$. In this paper two new 16 dimensional lattices with the center density $\{1}{16}$ and the kissing numbers 4224 and 4176 are constructed. This leads to several new 14 and 15 dimensional lattices which have the same center densities but different kissing numbers as the presently known densest lattices in these two dimensions. This gives a negative answer to the long time expectation that ${\bf \Lambda}_n$'s, $n \leq 24, n \neq 11,12,13$ are the only densest lattices in their dimensions. {abstract}