Lattice packings through division algebras

TL;DR

This paper introduces a novel method for constructing lattice packings in certain dimensions by generalizing previous results using division algebras, leading to improved bounds over existing techniques.

Contribution

It develops a new construction of lattice packings via division algebras and extends Siegel's mean value theorem to symmetric lattices, enhancing packing bounds.

Findings

Improved lattice packing bounds in many dimensions

Generalization of Venkatesh's lattice packing results

Development of an analogue of Siegel's mean value theorem for division algebra lattices

Abstract

In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensions.