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Designs, Codes and Cryptography

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01-01-2015

Symmetric doubly dual hyperovals have an odd rank

Author: Ulrich Dempwolff

Published in: Designs, Codes and Cryptography | Issue 1/2015

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Abstract

We show that a symmetric, doubly dual hyperoval has an odd rank. This is a weak support for the conjecture that doubly dual hyperovals over \(\mathbb{F }_2\) only exist, if the rank of the dual hyperoval is odd (see [2]).

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More common is notion \((n-1)\)-dimensional dual hyperoval for a dual hyperoval of rank \(n\) referring to the projective dimension of vectorspaces (see [5, Def. 2.1, 2.3]).

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Dempwolff U.: Dimensional doubly dual hyperovals and bent functions. Innov. Incid. Geom. (to appear).

Edel Y.: On some representations of quadratic APN functions and dimensional dual hyperovals. RIMS Kôkyûroku 1687, 118–130 (2010).

Taniguchi H.: On the duals of certain \(d\)-dimensional dual hyperovals in \(PG(2d+1,2)\). Finite Fields Appl. 15, 673–681 (2009).

Yoshiara S.: Dimensional dual arcs. In: Finite Geometries, Groups and Computation, Proceedings of Conference, Pingree Park 2004, Col. USA, pp. 247–266 (2006).

Title: Symmetric doubly dual hyperovals have an odd rank
Author: Ulrich Dempwolff
Publication date: 01-01-2015
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 1/2015
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-013-9847-y

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