On lattices generated by finite Abelian groups

TL;DR

This paper investigates lattices generated by finite Abelian groups, revealing their basis properties, improving bounds on covering radii for Barnes lattices, and analyzing automorphism groups, with implications for elliptic curves over finite fields.

Contribution

It demonstrates that most such lattices have bases of minimal vectors and improves existing bounds on their covering radii, expanding understanding of their structure.

Findings

Most lattices have bases of minimal vectors (except cyclic groups of order four)

Improved the upper bound for the covering radius of Barnes lattices

Analyzed automorphism groups of these lattices

Abstract

This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In case the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by Min Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.