Integral point sets over $\mathbb{Z}_n^m$

There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of $\mathbb{Z} / \mathbb{Z}n$, and study the properties of the resulting combinatorial structures.