Integral point sets in higher dimensional affine spaces over finite fields

Abstract

We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one of integral distances in Euclidean spaces. Therefore we expect the results over finite fields to be useful for the Euclidean case. We completely determine the automorphism group of these spaces which preserves integral distances. For some small parameters $m$ and $q$ we determine the maximum cardinality $\mathcal{I}(m,q)$ of integral point sets in $\mathbb{F}_q^m$. We provide upper bounds and lower bounds on $\mathcal{I}(m,q)$. If we map integral distances to edges in a graph, we can define a graph $\mathfrak{G}_{m,q}$ with vertex set $\mathbb{F}_q^m$. It turns out that $\mathfrak{G}_{m,q}$ is strongly regular for some cases.