Unimodular Hunting

TL;DR

This paper introduces a classification method for integral unimodular Euclidean lattices based on Kneser neighbors, successfully determining isometry classes for ranks 26 and 27.

Contribution

It develops a novel approach to classify unimodular lattices using Kneser neighbors, extending previous work and providing new classifications for higher ranks.

Findings

Classified unimodular lattices of rank 26 and 27.

Extended the classification framework for unimodular lattices.

Provided computational results for specific lattice ranks.

Abstract

We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n, which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.