Maximal integral point sets in affine planes over finite fields

TL;DR

This paper studies integral point sets in affine planes over finite fields, characterizing their structure, automorphisms, and maximal configurations, including classifications for small fields and new infinite series of maximal sets.

Contribution

It introduces new classifications of maximal integral point sets over finite fields, explores their automorphisms, and connects them to graph-theoretic structures like Paley graphs.

Findings

Classified maximal integral point sets for q ≤ 47

Identified two infinite series of maximal integral point sets

Established automorphism properties of these point sets

Abstract

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 $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\mathbb{F}_q$. It turns out that integral point sets over $\mathbb{F}_q$ 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 $\mathbb{F}_q$ for $q\le 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.