## 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 \$GF(q)^2\$ where the formally defined Euclidean distance of every pair of points is an element of \$GF(q)\$. From another point of view we consider point sets over \$GF(q)^2\$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integrWe consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in \$GF(q)^2\$ where the formally defined Euclidean distance of every pair of points is an element of \$GF(q)\$. From another point of view we consider point sets over \$GF(q)^2\$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over \$GF(q)^2\$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.
Wir betrachten ganzzahlige Punktmengen in affinen Ebenen über endlichen Körpern. Eine ganzzahlige Punktmenge ist hier eine Teilmenge von GF(q)^2, bei der der formal definierte Euklidische Abstand zwischen je zwei Punkten im Körper GF(q) liegt. In diesem Artikel betrachten wir das Spektrum der Kardinalitäten von inklusionsmaximalen ganzzahligen Punktmengen über GF(q)^2.