Integral point sets over finite fields

2008/04/08 by Sascha Kurz, Kurz, Sascha · 2 citations
Computer Science · Mathematics · #51E20 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #math.CO #msc:51E20

paper · pdf · doi:10.48550/arxiv.0804.1289

22 pages, 4 figures

arxiv created 2008/04/08 · openalex publication_date 2008/04/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider point sets in the affine plane \mathbbF_q² where each Euclidean distance of two points is an element of \mathbbF_q. These sets are called integral point sets and were originally defined in m-dimensional Euclidean spaces 𝔼^m. We determine their maximal cardinality I(\mathbbF_q,2). For arbitrary commutative rings R instead of \mathbbF_q or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.

Citations

• On chromatic number of unit-quadrance graphs (finite Euclidean graphs)
• On the minimum diameter of plane integral point sets

Cited by

• Integral point sets over ℤ_n^m
• Strongly regular graphs from integral point sets in even dimensional affine spaces over finite fields

Related

• Maximal integral point sets over ℤ²
• Integral point sets over ℤ_n^m
• On particular examples of planar integral point sets and their\n classification
• On the characteristic and diameter of planar integral point sets
• On existence of integral point sets and their diameter bounds