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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
Article
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

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).

Keywords

Dual hyperoval Presemifield Bilinear dual hyperoval 

Mathematics Subject Classification

05B25 51A45 51E20 51E21 

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 26400029.

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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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