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Singer 8-arcs of Mathon type in PG(2, 27)

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Abstract

De Clerck et al. (J Comb Theory, 2011) counted the number of non-isomorphic Mathon maximal arcs of degree-8 in PG(2, 2h), h ≠ 7 and prime. In this article we will show that in PG(2, 27) a special class of Mathon maximal arcs of degree-8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then count the total number of non-isomorphic Mathon maximal arcs of degree-8. Finally we show that the special arcs found in PG(2, 27) extend to two infinite families of Mathon arcs of degree-8 in PG(2, 2k), k odd and divisible by 7, while maintaining the nice property of admitting a Singer group.

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Authors and Affiliations

  1. Department of Mathematics, Ghent University, Krijgslaan 281–S22, 9000, Ghent, Belgium

    Frank De Clerck & Thomas Maes

  2. Department of Mathematics, Morton Hall 321, 1 Ohio University, Athens, OH, 45701, USA

    Stefaan De Winter

Authors
  1. Frank De Clerck
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  2. Stefaan De Winter
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  3. Thomas Maes
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Corresponding author

Correspondence to Frank De Clerck.

Additional information

Stefaan De Winter is Postdoctoral Fellow of the Science Foundation—Flanders.

This is one of several papers published together in “Designs, Codes and Cryptography” on the special topic: Geometry, Combinatorial Designs & Cryptology.

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De Clerck, F., De Winter, S. & Maes, T. Singer 8-arcs of Mathon type in PG(2, 27). Des. Codes Cryptogr. 64, 17–31 (2012). https://doi.org/10.1007/s10623-011-9502-4

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  • DOI: https://doi.org/10.1007/s10623-011-9502-4

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