Inclusion-maximal integral point sets over finite fields

We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in F2q where the formally defined Euclidean distance of every pair of points is an element of Fq. From another point of view we consider point sets over F2q with few and prescribed directions. So this is related to Rédei’s work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces Em. In this article we study the spectrum of integral point sets over Fq which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.