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Published in: Designs, Codes and Cryptography 2/2016

01-08-2016

Dimensional dual hyperovals in classical polar spaces

Author: John Sheekey

Published in: Designs, Codes and Cryptography | Issue 2/2016

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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.
Literature
1.
go back to reference Brouwer A.E., Cohen A.M., Neumaier A.: Distance Regular Graphs. Springer, New York (1989). Brouwer A.E., Cohen A.M., Neumaier A.: Distance Regular Graphs. Springer, New York (1989).
2.
go back to reference 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). 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).
3.
go back to reference Del Fra A.: On d-dimensional dual hyperovals. Geom. Dedicata 79, 157–178 (2000). Del Fra A.: On d-dimensional dual hyperovals. Geom. Dedicata 79, 157–178 (2000).
4.
go back to reference Dempwolff U.: Dimensional doubly dual hyperovals and bent functions. Innov. Incid. Geom. 13, 149–178 (2013). Dempwolff U.: Dimensional doubly dual hyperovals and bent functions. Innov. Incid. Geom. 13, 149–178 (2013).
5.
go back to reference Dempwolff U.: Symmetric doubly dual hyperovals have an odd rank. Des. Codes Cryptogr. 74, 153–157 (2015). Dempwolff U.: Symmetric doubly dual hyperovals have an odd rank. Des. Codes Cryptogr. 74, 153–157 (2015).
6.
go back to reference Dempwolff U., Kantor W.M.: Orthogonal dual hyperovals, symplectic spreads and orthogonal spreads. J. Algebr. Comb. 41, 83–108 (2015). Dempwolff U., Kantor W.M.: Orthogonal dual hyperovals, symplectic spreads and orthogonal spreads. J. Algebr. Comb. 41, 83–108 (2015).
7.
go back to reference Edel Y.: On some representations of quadratic APN functions and dimensional dual hyperovals. RIMS Kokyuroku 1687, 118–130 (2010). Edel Y.: On some representations of quadratic APN functions and dimensional dual hyperovals. RIMS Kokyuroku 1687, 118–130 (2010).
8.
go back to reference 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). 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).
9.
go back to reference 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). 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).
10.
go back to reference Taniguchi H.: On the duals of certain d-dimensional dual hyperovals in \({\text{ PG }}(2d+1,2)\). Finite Fields Appl. 15, 673–681 (2009). Taniguchi H.: On the duals of certain d-dimensional dual hyperovals in \({\text{ PG }}(2d+1,2)\). Finite Fields Appl. 15, 673–681 (2009).
11.
go back to reference 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). 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).
12.
go back to reference Vanhove F.: Incidence geometry from an algebraic graph theory point of view. Ph.D. Thesis (2011a). Vanhove F.: Incidence geometry from an algebraic graph theory point of view. Ph.D. Thesis (2011a).
13.
go back to reference Vanhove F.: Antidesigns and regularity of partial spreads in dual polar graphs. J. Comb. Des. 19, 202–216 (2011b). Vanhove F.: Antidesigns and regularity of partial spreads in dual polar graphs. J. Comb. Des. 19, 202–216 (2011b).
14.
go back to reference Yoshiara S.: A family of d-dimensional dual hyperovals in \({\text{ PG }}(2d + 1, 2)\). Eur. J. Comb. 20, 589–603 (1999). Yoshiara S.: A family of d-dimensional dual hyperovals in \({\text{ PG }}(2d + 1, 2)\). Eur. J. Comb. 20, 589–603 (1999).
15.
go back to reference Yoshiara S.: Some remarks on dimensional dual hyperovals of polar type. Bull. Belg. Math. Soc. Simon Stevin 12, 925–939 (2005). Yoshiara S.: Some remarks on dimensional dual hyperovals of polar type. Bull. Belg. Math. Soc. Simon Stevin 12, 925–939 (2005).
16.
go back to reference Yoshiara S.: Dimensional dual arcs: a survey. In: Finite Geometries, Groups, and Computation, pp. 247–266. Walter de Gruyter GmbH & Co. KG, Berlin (2006). Yoshiara S.: Dimensional dual arcs: a survey. In: Finite Geometries, Groups, and Computation, pp. 247–266. Walter de Gruyter GmbH & Co. KG, Berlin (2006).
Metadata
Title
Dimensional dual hyperovals in classical polar spaces
Author
John Sheekey
Publication date
01-08-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-015-0105-3

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