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Designs, Codes and Cryptography

, Volume 80, Issue 2, pp 415–420 | Cite as

Dimensional dual hyperovals in classical polar spaces

  • John SheekeyEmail author
Article
  • 159 Downloads

Abstract

In this paper we show that n-dimensional dual hyperovals cannot exist in all but one classical polar space of rank n if n is even. This resolves a question posed by Yoshiara.

Keywords

Dimensional dual hyperoval Dual polar graph Polar space Dimensional dual arc 

Mathematics Subject Classification

51A50 51E21 

Notes

Acknowledgments

The results of this paper were developed in discussions with Frèdèric Vanhove prior to his tragic early passing. The author is heavily indebted to Frèdèric for this work, and this paper is dedicated to his memory. The author is supported by the Research Foundation—Flanders (FWO—Vlaanderen).

References

  1. 1.
    Brouwer A.E., Cohen A.M., Neumaier A.: Distance Regular Graphs. Springer, New York (1989).Google Scholar
  2. 2.
    De Beule J., Klein A., Metsch K.: Current research topics in Galois geometry. In: J De Beule J., Storme L. (eds.) Substructures of Finite Classical Polar Spaces. NOVA Academic Publishers, New York (2011).Google Scholar
  3. 3.
    Del Fra A.: On d-dimensional dual hyperovals. Geom. Dedicata 79, 157–178 (2000).Google Scholar
  4. 4.
    Dempwolff U.: Dimensional doubly dual hyperovals and bent functions. Innov. Incid. Geom. 13, 149–178 (2013).Google Scholar
  5. 5.
    Dempwolff U.: Symmetric doubly dual hyperovals have an odd rank. Des. Codes Cryptogr. 74, 153–157 (2015).Google Scholar
  6. 6.
    Dempwolff U., Kantor W.M.: Orthogonal dual hyperovals, symplectic spreads and orthogonal spreads. J. Algebr. Comb. 41, 83–108 (2015).Google Scholar
  7. 7.
    Edel Y.: On some representations of quadratic APN functions and dimensional dual hyperovals. RIMS Kokyuroku 1687, 118–130 (2010).Google Scholar
  8. 8.
    Gow R., Lavrauw M., Sheekey J., Vanhove F.: Constant rank-distance sets of Hermitian matrices and partial spreads in Hermitian polar spaces, Electron. J. Comb. 21, Paper 1.26, 19(2014).Google Scholar
  9. 9.
    Ihringer F.: A new upper bound for constant distance codes of generators on Hermitian polar spaces of type \(H(2d--1, q^2)\). J. Geom. 105, 457–464 (2014).Google Scholar
  10. 10.
    Taniguchi H.: On the duals of certain d-dimensional dual hyperovals in \({\text{ PG }}(2d+1,2)\). Finite Fields Appl. 15, 673–681 (2009).Google Scholar
  11. 11.
    Vanhove F.: The maximum size of a partial spread in \(H(4n + 1, q^2)\) is \(q^{2n+1} + 1\). Electron. J. Comb. 16, 1–6 (2009).Google Scholar
  12. 12.
    Vanhove F.: Incidence geometry from an algebraic graph theory point of view. Ph.D. Thesis (2011a).Google Scholar
  13. 13.
    Vanhove F.: Antidesigns and regularity of partial spreads in dual polar graphs. J. Comb. Des. 19, 202–216 (2011b).Google Scholar
  14. 14.
    Yoshiara S.: A family of d-dimensional dual hyperovals in \({\text{ PG }}(2d + 1, 2)\). Eur. J. Comb. 20, 589–603 (1999).Google Scholar
  15. 15.
    Yoshiara S.: Some remarks on dimensional dual hyperovals of polar type. Bull. Belg. Math. Soc. Simon Stevin 12, 925–939 (2005).Google Scholar
  16. 16.
    Yoshiara S.: Dimensional dual arcs: a survey. In: Finite Geometries, Groups, and Computation, pp. 247–266. Walter de Gruyter GmbH & Co. KG, Berlin (2006).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Universiteit GentGhentBelgium

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