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Study on higher dimensional dual hyperovals and related functions on finite fields

Research Project

Project/Area Number 26400029
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionKagawa National College of Technology

Principal Investigator

Taniguchi Hiroaki  香川高等専門学校, 一般教育科, 教授 (60370037)

Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords代数学 / 代数的組合せ論 / 有限幾何学 / 高次元双対超卵形 / APN関数 / 半体 / 有限射影平面 / 有限射影空間
Outline of Final Research Achievements

We discovered a family of dual hyperovals S_c(l,GF(2r)), where l, r integers and c an element of the field GF(2r). Using these dual hyperovals S_c(l,GF(2r)), we construct many simply connected examples which are not known before. We describe the cover and quotient relations among them using elements c and the integers l and r, and determined the automorphism groups of them. Next we construct dual hyperovals using a commutative presemified and determined the isomorphism problems among them. By extending this idea, we also construct dual hyperovals using many presemifields which may not be commutative. We determined the isomorphism problems among them under the specific conditions, such as the presemifields are not isotopic to commutative presemifields. Lastly, we construct a quadratic APN function using a bent function of special type. Only two examples of such constructions were known before. It is proved that our construction are not equivalent to the former two examples.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (22 results)

All 2018 2017 2016 2015 2014

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Acknowledgement Compliant: 4 results) Presentation (17 results) (of which Int'l Joint Research: 6 results,  Invited: 1 results)

  • [Journal Article] Bilinear dual hyperovals from binary commutative presemifields II2018

    • Author(s)
      Hiroaki Taniguchi
    • Journal Title

      Finite Fields and Their Applications

      Volume: 49 Pages: 42-79

    • DOI

      10.1016/j.ffa.2017.08.009

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On some bilinear dual hyperovals2017

    • Author(s)
      Hiroaki Taniguchi
    • Journal Title

      Discrete Mathematics

      Volume: 340 Issue: 1 Pages: 3154-3166

    • DOI

      10.1016/j.disc.2016.07.002

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Bilinear dual hyperovals from binary commutative presemifields2016

    • Author(s)
      Hiroaki Taniguchi
    • Journal Title

      Finite Fields and Their Applications

      Volume: 42 Pages: 93-101

    • DOI

      10.1016/j.ffa.2016.07.003

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Some examples of simply connected dual hyperovals II2015

    • Author(s)
      Hiroaki Taniguchi
    • Journal Title

      Finite Fields and Their Applications

      Volume: 36 Pages: 1-13

    • DOI

      10.1016/j.ffa.2015.06.002

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] New dimensional dual hyperovals, which are not quotients of the classical dual hyperovals2014

    • Author(s)
      Hiroaki Taniguchi
    • Journal Title

      Discrete Mathematics

      Volume: 337 Pages: 65-75

    • DOI

      10.1016/j.disc.2014.08.004

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] ある2次的なAPN関数について2018

    • Author(s)
      谷口浩朗
    • Organizer
      研究集会「有限幾何とその周辺」
    • Related Report
      2017 Annual Research Report
  • [Presentation] Bilinearでない高次元双対超卵形の例2017

    • Author(s)
      谷口浩朗
    • Organizer
      有限幾何学とその周辺
    • Place of Presentation
      東京女子大学
    • Year and Date
      2017-03-24
    • Related Report
      2016 Research-status Report
  • [Presentation] Dual Hyperovals from Commutative Presemifields2017

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      48th Southeastern International Conference on Combinatorics, Graph Theory & Computing
    • Place of Presentation
      Florida Atlantic University, Boca Raton, USA
    • Year and Date
      2017-03-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Dual hyperovals from three or more presemifields2017

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      The 13th International Conference on Finite Fields and their Applications
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] A variation of the dual hyperoval S_c using presemifields2017

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      Finite Geometries Fifth Irsee Conference
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] あるBilinear DHOの変形2017

    • Author(s)
      谷口浩朗
    • Organizer
      研究集会「有限幾何とその周辺」
    • Related Report
      2017 Annual Research Report
  • [Presentation] 3つの半体から構成されるdual hyperovalについて,2016

    • Author(s)
      谷口浩朗
    • Organizer
      有限幾何学とその周辺
    • Place of Presentation
      東京女子大学
    • Year and Date
      2016-09-04
    • Related Report
      2016 Research-status Report
  • [Presentation] Bilinear dual hyperovals from binary commutative presemifields2016

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      Combinatorics 2016
    • Place of Presentation
      Maratea, Italy
    • Year and Date
      2016-05-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] 半体と双対超卵形2016

    • Author(s)
      谷口浩朗
    • Organizer
      「有限幾何とその周辺-平峰先生を偲んで」研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2016-03-05
    • Related Report
      2015 Research-status Report
  • [Presentation] On some bilinear dual hyperovals2015

    • Author(s)
      谷口浩朗
    • Organizer
      「有限幾何とその周辺」研究集会
    • Place of Presentation
      東京女子大学
    • Year and Date
      2015-09-26
    • Related Report
      2015 Research-status Report
  • [Presentation] On some bilinear dual hyperovals2015

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      Giornate di Giometria
    • Place of Presentation
      Caserta (Italy)
    • Year and Date
      2015-09-17
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] On some dual hyperovals2015

    • Author(s)
      HIroaki Taniguchi
    • Organizer
      Fq12
    • Place of Presentation
      Saratoga Springs (USA)
    • Year and Date
      2015-07-12
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 高次元双対超卵形について2015

    • Author(s)
      谷口浩朗
    • Organizer
      第32回代数的組み合わせ論シンポジウム
    • Place of Presentation
      金沢大学
    • Year and Date
      2015-06-22
    • Related Report
      2015 Research-status Report
  • [Presentation] 単連結な高次元dual hyperovalの例2015

    • Author(s)
      谷口浩朗
    • Organizer
      平成 26 年度 日本数学会 中国・四国支部例会
    • Place of Presentation
      徳島大学
    • Year and Date
      2015-01-25
    • Related Report
      2014 Research-status Report
  • [Presentation] Some examples of simply connected dual hyperovals2015

    • Author(s)
      谷口浩朗
    • Organizer
      熊本組合せ論研究集会–代数的デザイン論とその周辺–
    • Place of Presentation
      熊本大学
    • Year and Date
      2015-01-09 – 2015-01-11
    • Related Report
      2014 Research-status Report
  • [Presentation] 高次元のdual hyperovalについて2014

    • Author(s)
      谷口浩朗
    • Organizer
      Workshop on Galois point and related topics
    • Place of Presentation
      滋賀大学大津サテライトプラザ
    • Year and Date
      2014-09-13 – 2014-09-15
    • Related Report
      2014 Research-status Report
  • [Presentation] On Covering maps of Bilinear Dual Hyperovals2014

    • Author(s)
      Hiroaki Taniguchi
    • Organizer
      Combinatorics 2014
    • Place of Presentation
      Gaeta, Italy
    • Year and Date
      2014-07-01 – 2014-07-06
    • Related Report
      2014 Research-status Report

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Published: 2014-04-04   Modified: 2019-03-29  

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