  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.
