Long shortest vectors in three dimensional lattices

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 $N$. As a main result, we present an infinite family of lattices whose appropriately normalised shortest vectors converge to a heuristic upper bound based on the optimal lattice packing in $\mathbb{R}^3$.