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- Publisher
- Springer Journals
- Copyright
- Copyright © János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2022
- ISSN
- 0209-9683
- eISSN
- 1439-6912
- DOI
- 10.1007/s00493-021-4712-5
- Publisher site
- See Article on Publisher Site
Abstract
Let m be a positive integer, q be a prime power, and PG(2,q) be the projective plane over the finite field Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{F}_q}$$\end{document}. Finding complete m-arcs in PG(2,q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any m ≥ 8. For any fixed m, our arcs Aq,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\cal A}_{q,m}}$$\end{document} satisfy |Aq,m|−q→−∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left| {{{\cal A}_{q,m}}} \right| - q \to - \infty $$\end{document} as q grows. To produce such m-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the m-completeness of the arc.
Journal
Combinatorica – Springer Journals
Published: Oct 1, 2022
Keywords: 05B25; 51E21; 11R45; 51E20