Inclusion-maximal integral point sets over finite fields

Details Inclusion-maximal integral point sets over finite fields Inclusion-maximal integral point sets over finite fields

2008-04-08

arxiv.org/abs/0804.1285v1

Mathematics

Combinatorics

15 pages, 3 figures

Scientific paper

We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $\mathbb{F}_q^2$ where the formally defined Euclidean distance of every pair of points is an element of $\mathbb{F}_q$. From another point of view we consider point sets over $\mathbb{F}_q^2$ with few and prescribed directions. So this is related to R\'edei's work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces $\mathbb{E}^m$. In this article we study the spectrum of integral point sets over $\mathbb{F}_q$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.

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