On the construction of dense lattices with a given automorphism group

TL;DR

This paper presents a method to construct dense lattices in high dimensions with specified automorphism groups, achieving near-optimal packing densities and improving the complexity of lattice basis construction.

Contribution

It introduces a new construction technique for dense lattices with prescribed automorphism groups using double circulant codes, matching known density bounds.

Findings

Lattices with density at least cn/2^n are constructed.

A family of lattices with automorphism group size n achieves the density bound.

The algorithm for basis construction runs in exponential time, improving previous results.

Abstract

We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions n, we exhibit a finite set of lattices that come with an automorphism group of size n, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp(nlogn), which improves upon previous theorems that yield an equivalent lattice packing density. The method developed here involves applying Leech and Sloane's construction A to a special class of codes with a given automorphism group, namely the class of double circulant codes.