Bases of minimal vectors in lattices, III

TL;DR

This paper investigates the properties of minimal vector bases in Euclidean lattices, proving their existence in dimensions up to 9 and providing a counterexample for higher dimensions.

Contribution

It establishes the existence of minimal vector bases in low dimensions and presents a counterexample demonstrating the limitation in higher dimensions.

Findings

All lattices of dimension ≤ 9 generated by minimal vectors have a minimal basis.

Counterexample shows not all higher-dimensional lattices have minimal vector bases.

The result clarifies the dimensional boundary for minimal basis existence.

Abstract

We prove that all Euclidean lattices of dimension $n\le 9$ which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions $n\ge 10$.