The study on higher dimensional dual hyperovals in finite geometry

• Project/Area Number: 20540034
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Takuma National College of Technology
• Principal Investigator: TANIGUCHI Hiroaki Takuma National College of Technology, 一般教育科, 教授 (60370037)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2008: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 代数的組合せ論 / 代数学 / 有限幾何学 / 高次元双対超卵形 / APN関数 / 有限射影空間

Research Abstract

We study on the relation between APN (Almost Perfect Non-linear) functions, which are investigated in the field of cryptography, and dual hyperovals. Using this relation, we construct new dual hyperovals. Next, using semifields, we construct many new dual hyperovals. Moreover, we also construct many new non isomorphic quotients of the Veronesean dual hyperoval, and the Buratti Del Fra dual hyperoval in small dimensional projective spaces.

Research Products

• [Journal Article] On $d$-dimensional Buratti-Del Fra type dual hyperovals in $PG(3d,2)$ (2010)
  • Author(s): 谷口浩朗
  • Journal Title: Discrete Mathematics Vol.310 Pages: 3633-3645
  • Peer Reviewed
• [Journal Article] On d-dimensional Buratti-Del Fra type dual hyperovals in PG(3d, 2) (2010)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: Discrete Mathematics Volume: 310 Pages: 3633-3645
  • Peer Reviewed
• [Journal Article] On some d-dimensional dual hyperovals in PG(d(d+1)/2,2) (2009)
  • Author(s): 谷口浩朗
  • Journal Title: European Journal of Combinatorics Vol.31 Pages: 401-410
  • Peer Reviewed
• [Journal Article] On d-dimensional dual hyperovals in PG(2d,2) (2009)
  • Author(s): 谷口浩朗
  • Journal Title: Innovations in Incidence Geometry Vol.8 Pages: 137-145
  • Peer Reviewed
• [Journal Article] On the duals of certain dual hyperovals in PG(2d+1、2) (2009)
  • Author(s): 谷口浩朗
  • Journal Title: Finite Fields and Their Applications Vol.15 Pages: 673-681
  • Peer Reviewed
• [Journal Article] On d-dimensional dual hyperovals in PG(2d, 2) (2009)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: Innovations in Incidence Geometry 8 Pages: 137-145
  • Peer Reviewed
• [Journal Article] On the duals of certain dual hyperovals in PG(2d+1, 2) (2009)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: Finite Fields and Their Applications 15 Pages: 673-681
  • Peer Reviewed
• [Journal Article] On some d-dimensional dual hyperovals in PG(d(d+1)/2, 2) (2009)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: European Journal of Combinatorics 31 Pages: 401-410
  • Peer Reviewed
• [Journal Article] A new family of dual hyperovals in PG (d(d+3)/2, 2) with d≧3 (2009)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: Discrete Mathematics 309 Pages: 418-429
  • Peer Reviewed
• [Journal Article] On some d-dimensional dual hyperovals in PG(2d、2) (2008)
  • Author(s): 谷口浩朗
  • Journal Title: Finite Fields and Their Applications Vol.14 Pages: 1010-1019
  • Peer Reviewed
• [Journal Article] On automorphism groups of some d-dimensional dual hyperovals in PG (d(d+3)/2, 2) (2008)
  • Author(s): Hiroaki taniguchi
  • Journal Title: Graphs and Combinatorics 24 Pages: 229-236
  • Peer Reviewed
• [Journal Article] On some d-dimensional dual hyperovals in PG (2d, 2) (2008)
  • Author(s): Hiroaki Taniguchi
  • Journal Title: Finite Fields and Their Applications 14 Pages: 1010-1019
  • Peer Reviewed
• [Presentation] 有限幾何とその周辺 (2010)
  • Author(s): 谷口浩朗
  • Organizer: On dimensional dual hyperovals研究集会
  • Place of Presentation: 熊本大学
  • Year and Date: 2010-11-13
• [Presentation] On the quotients of dual hyperovals (2010)
  • Author(s): Hiroaki Taniguchi
  • Organizer: Combinatorics 2010
  • Place of Presentation: Verbania, Italy
  • Year and Date: 2010-07-19
• [Presentation] (2010)
  • Author(s): 谷口浩朗
  • Organizer: On the quotients of dual hyperovals Combinatorics2010
  • Place of Presentation: ヴェルバニア、イタリア
  • Year and Date: 2010-07-01
• [Presentation] (2010)
  • Author(s): 谷口浩朗
  • Organizer: On some quotients of dual hyperovals in PG(d(d+3)/2、2)第27回代数的組合せ論シンポジウム
  • Place of Presentation: 高知大学
  • Year and Date: 2010-06-22
• [Presentation] On some quotients of dual hyperovals in PG(d(d+3)/2,2) (2010)
  • Author(s): 谷口浩朗
  • Organizer: 第27回代数的組合せ論シンポジウム
  • Place of Presentation: 高知大学
  • Year and Date: 2010-06-22
• [Presentation] 代数的組合せ論 (2010)
  • Author(s): 谷口浩朗
  • Organizer: d-dimensional dual hyperovals in $PG(d(d+3)/2、2)$ and their quotientsミニ集会
  • Place of Presentation: 神戸学院大学
  • Year and Date: 2010-03-18
• [Presentation] Buratti-Del Fra型の高次元双対超卵形について (2009)
  • Author(s): 谷口浩朗
  • Organizer: 数理解析研究所共同事業「代数的組合せ論および群と代数研究集会」
  • Place of Presentation: 信州大学
  • Year and Date: 2009-11-18
• [Presentation] Buratti-Del Fra型の高次元双対超卵形について数理解析研究所 共同研究事業 (2009)
  • Author(s): 谷口浩朗
  • Organizer: 代数的組合せ論および群と代数研究集会
  • Place of Presentation: 信州大学
  • Year and Date: 2009-11-16
• [Presentation] 有限幾何とその周辺 (2009)
  • Author(s): 谷口浩朗
  • Organizer: Dual of some dual hyperovals研究集会
  • Place of Presentation: 近畿大学
  • Year and Date: 2009-10-17
• [Presentation] 有限幾何とその周辺 (2009)
  • Author(s): 谷口浩朗
  • Organizer: あるAPN関数から構成されるDHOのdual について 研究集会
  • Place of Presentation: 東京女子大学
  • Year and Date: 2009-08-07
• [Presentation] 有限幾何とその周辺 (2009)
  • Author(s): 谷口浩朗
  • Organizer: 3D-Cubeと高次元双対超卵形 研究集会
  • Place of Presentation: 東海大学阿蘇キャンパス
  • Year and Date: 2009-05-16
• [Presentation] 有限幾何とその周辺 (2009)
  • Author(s): 谷口浩朗
  • Organizer: 概体と双対超卵形 研究集会
  • Place of Presentation: 福岡大学
  • Year and Date: 2009-01-24
• [Presentation] 概体と双対超卵形 (2009)
  • Author(s): 谷口浩朗
  • Organizer: 有限幾何とその周辺
  • Place of Presentation: 福岡大学
  • Year and Date: 2009-01-24