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Advances in Geometry

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Maximal arcs in projective planes of order 16 and related designs

Mustafa Gezek / Vladimir D. Tonchev
  • Corresponding author
  • Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA
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  • Other articles by this author:
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/ Tim Wagner
  • Corresponding author
  • Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2018-0002

Abstract

The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

Keywords: Maximal arc; projective plane; resolvable design

MSC 2010: 05B05; 51E10

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About the article


Received: 2017-07-28

Published Online: 2018-03-26


Funding: Research supported by NSA Grant H98230-16-1-0011.


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0002.

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