Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 895–908 | Cite as

A variation of the dual hyperoval \({\mathcal {S}}_c\) using presemifields

  • Hiroaki TaniguchiEmail author
Part of the following topical collections:
  1. Special Issue: Finite Geometries


In Discret Math 337:65–75, 2014, we construct a bilinear dual hyperoval called \({\mathcal {S}}_c(l,GF(2^r))\), or simply \({\mathcal {S}}_c\), for \(rl\ge 4\) and \(c\in GF(2^r)\) with \(Tr(c)=1\). In this note, we modify the bilinear mapping of \({\mathcal {S}}_c\) for \(l \ge 2\) using multiplications of presemifields, and have a dual hyperoval \({\mathcal {S}}_{c}^{'}\) from this bilinear mapping. We also investigate on the isomorphism problems of these dual hyperovals under the conditions that \(c\ne 1\) and the presemifields are not isotopic to commutative presemifields (see Theorem 2 for precise statement), and see that, under these conditions, \({\mathcal {S}}_{c}^{'}\) is not isomorphic to the dual hyperovals in Taniguchi (Discret Math 337:65–75, 2014).


Dual hyperoval Presemifield Bilinear dual hyperoval 

Mathematics Subject Classification

05B25 51A45 51E20 51E21 



This work was supported by JSPS KAKENHI Grant Number 26400029.


  1. 1.
    Albert A.A.: Generalized twisted fields. Pac. J. Math. 8, 1–8 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biliotti M., Jha V., Johnson N.: The collineation groups of generalized twisted field planes. Geom. Dedic. 76, 97–126 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dempwolff U., Edel Y.: Dimensional dual hyperovals and APN functions with translation groups. J. Comb. 39, 457–496 (2014).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Huybrechts C., Pasini A.: Flag transitive extensions of dual affine spaces. Contrib. Algebr. Geom. 40, 503–532 (1999).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kantor W.M.: Finite semifields. In: Finite Geometries, Groups, and Computation, (Proceedings of Conference at Pingree Park, CO Sept. 2005). Walter de Gruyter, Berlin, pp. 103–114 (2006).Google Scholar
  6. 6.
    Kantor W.M.: Commutative semifields and symplectic spreads. J. Algebr. 270(1), 96–114 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnson N., Jha V., Biliotti M.: Handbook of Finite Translation Planes. Chapman & Hall/CRC, Boca Raton (2007).zbMATHGoogle Scholar
  8. 8.
    Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, pp. 131–160. Nova Science Publishers, Inc., New York (2011).Google Scholar
  9. 9.
    Taniguchi H.: New dimensional dual hyperovals, which are not quotients of the classical dual hyperovals. Discret. Math. 337, 65–75 (2014).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Institute of Technology, Kagawa CollegeTakamatsuJapan

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