Even Unimodular Lattices from Quaternion Algebras

TL;DR

This paper explores a quaternion algebra-based lattice construction method that simplifies the process of obtaining high-dimensional extremal and dense lattices, including explicit constructions of well-known lattices like E8 and Lambda16.

Contribution

It introduces a novel quaternion algebra approach over number fields that reduces problem dimensionality and enables explicit constructions of key lattices in dimensions 8 and 16.

Findings

Constructed E8, E8^2, and Lambda16 lattices from quaternion algebras over number fields.

Provided density results for such number fields.

Presented a method for constructing even unimodular lattices in dimensions multiple of 8.

Abstract

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and $\Lambda_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of $8$.