Long shortest vectors in low dimensional lattices

TL;DR

This paper investigates point sets generated by coprime integers modulo N, relating them to lattices, and constructs infinite families of lattices with shortest vectors approaching the Hermite constant in low dimensions.

Contribution

It introduces a new construction linking point sets modulo N to lattices and demonstrates infinite families with shortest vectors approaching the Hermite constant in 3, 4, and 5 dimensions.

Findings

Constructed infinite families of lattices with shortest vectors approaching the Hermite constant.

Established a relationship between point sets modulo N and specific lattice structures.

Extended results to dimensions 4 and 5.

Abstract

For coprime integers $N,a,b,c$, with $0<a<b<c<N$, we define the set $$ \{ (na \! \! \! \! \pmod{N}, nb \! \! \! \! \pmod{N}, nc \! \! \! \! \pmod{N}) : 0 \leq n < N\}. $$ We study which parameters $N,a,b,c$ generate point sets with long shortest distances between the points of the set in dependence of $N$ and relate such sets to lattices of a particular form. As a main result, we present an infinite family of such lattices with the property that the normalised norm of the shortest vector of each lattice converges to the square root of the Hermite constant $\gamma_3$. We obtain a similar result for the generalisation of our construction to $4$ and $5$ dimensions.