Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane F2q over a finite field Fq, where the formally defined squared Euclidean distance of every pair of points is a square in Fq. It turns out that integral point sets over Fq can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over Fq for q ≤ 47. Furthermore, we give two series of maximal integral point sets and prove their maximality.