A new lower bound for the smallest complete (k, n)-arc in PG(2,q) View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2018-12-21

AUTHORS

S. Alabdullah, J. W. P. Hirschfeld

ABSTRACT

In PG(2,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {PG}(2,q)$$\end{document}, the projective plane over the field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{F}_{q}$$\end{document} of q elements, a (k, n)-arc is a set K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} is complete, that is, not contained in a (k+1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1,n)$$\end{document}-arc. In particular, what are the smallest and largest values of k for a complete K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document}, denoted by tn(2,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_n(2,q)$$\end{document} and mn(2,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_n(2,q)$$\end{document}? Here, a new lower bound for tn(2,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_n(2,q)$$\end{document} is established and compared to known values for small q. More... »

PAGES

679-683

References to SciGraph publications

  • 2004-08. Classification of the (n, 3)-arcs in PG(2, 7) in JOURNAL OF GEOMETRY
  • 2015-04-12. Complete (k,3)-arcs from quartic curves in DESIGNS, CODES AND CRYPTOGRAPHY
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10623-018-00592-8

    DOI

    http://dx.doi.org/10.1007/s10623-018-00592-8

    DIMENSIONS

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