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Constructive research towards solving the existence problem on Hadamard matrices

Research Project

Project/Area Number 20K03719
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12030:Basic mathematics-related
Research InstitutionKumamoto University

Principal Investigator

Momihara Koji  熊本大学, 大学院先端科学研究部(理), 准教授 (70613305)

Co-Investigator(Kenkyū-buntansha) 丸田 辰哉  大阪公立大学, 理学(系)研究科(研究院), 教授 (80239152)
Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2021: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsアダマール行列 / 有限幾何 / 2-交差集合 / 差集合族 / ガウス和 / 強正則グラフ / アソシエーションスキーム / divisible design graph / 自己双対符号 / 歪みアダマール行列 / D-最適性 / 差集合 / 均整アダマール行列
Outline of Research at the Start

これまでの研究で,「群の作用」と「ガウス和の計算」を併用した高次元有限射影空間における超平面交差数の計算法を考案してきた. この手法を発展・応用し, 2-交差集合を用いたアダマール行列の既存の幾何学的構成法を高次元化することで, 平方数次均整アダマール行列を組織的に生成する新たな構成法を開発する.
また, 申請者の最近の研究で, アソシエーションスキームを通じ, (4×非平方数)次アダマール行列と均整アダマール行列との関係性が明らかになってきた. 本研究では, その関係に基づき, アソシエーションスキームの枠組みの中で, (4×非平方数)次アダマール行列を組織的に構成する手法・理論を確立する.

Outline of Final Research Achievements

The objective of this research project is to build a theory for generating new Hadamard matrices by generalizing known geometric constructions of Hadamard matrices using two-intersection sets in finite projective spaces and by giving new constructions of Hadamard matrices based strongly regular graphs and association schemes.
In this research, we found new methods to construct geometric objects using actions of finite groups and power residues of finite fields and gave new characterizations for Gauss sums and Gauss periods to compute hyperplane sections. In particular, we succeeded to find families of new Hadamard matrices. Furthermore, we proposed new constructions of strongly regular graphs and association schemes related to Hadamard matrices. Thus, we developed the existence theory of Hadamard matrices and related combinatorial objects.

Academic Significance and Societal Importance of the Research Achievements

アダマール行列は, 符号・デザイン・グラフ・格子等の構成に用いられる重要な離散構造であり, 新たなアダマール行列や関連する組合せ構造の発見が, これらの分野における様々な問題を解決する可能性もあり, 多くの付加的成果を生み出すという点で本研究成果には重要な意義がある. また, 有限幾何やガウス和をはじめとする整数論における研究成果は, 組合せ論における様々な構造の存在証明に応用できる潜在的可能性もあり, 重要な価値がある. 更には, アダマール行列は, 統計における分散分析・情報通信における周波数信号処理等, 情報科学分野に多くの応用があるため, 幅広く社会に貢献することが期待される.

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (25 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (4 results) Journal Article (13 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 12 results,  Open Access: 4 results) Presentation (5 results) (of which Int'l Joint Research: 2 results,  Invited: 3 results) Remarks (3 results)

  • [Int'l Joint Research] 南方科技大学(中国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] 南方科技大学(中国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of Delaware/Department of Mathematics(米国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] National University of Singapore/Division of Natural Science(シンガポール)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Strongly Regular Graphs from Pseudocyclic Association Schemes2024

    • Author(s)
      Momihara Koji、Suda Sho
    • Journal Title

      Graphs and Combinatorics

      Volume: 40 Issue: 2

    • DOI

      10.1007/s00373-024-02764-x

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Difference sets in pseudocyclic association schemes2023

    • Author(s)
      Kajiura Hiroki、Momihara Koji
    • Journal Title

      Discrete Mathematics

      Volume: 346 Issue: 12 Pages: 113598-113598

    • DOI

      10.1016/j.disc.2023.113598

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The existence of Cameron-Liebler line classes with parameter $\frac{(q+1)^2}{3}$2023

    • Author(s)
      籾原幸二
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2253

    • Related Report
      2023 Annual Research Report
  • [Journal Article] Hadamard matrices related to a certain series of ternary self-dual codes2023

    • Author(s)
      M. Araya, M. Harada, K. Momihara
    • Journal Title

      Designs, Codes and Cryptography

      Volume: 91 Issue: 3 Pages: 795-805

    • DOI

      10.1007/s10623-022-01127-y

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on avoidance games on Steiner triple systems2022

    • Author(s)
      N. Matsumoto, K. Momihara, M. Nakamura
    • Journal Title

      Australasian Journal of Combinatorics

      Volume: 83 Pages: 196-293

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Square (1,-1)-matrices with large determinants and near-Hadamard matrices2022

    • Author(s)
      K. Momihara, S. Suda, Q. Xiang
    • Journal Title

      Linear Algebra and its Applications

      Volume: 654 Pages: 28-55

    • DOI

      10.1016/j.laa.2022.08.028

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields2022

    • Author(s)
      Momihara Koji、Xiang Qing
    • Journal Title

      Finite Fields and Their Applications

      Volume: 77 Pages: 101953-101953

    • DOI

      10.1016/j.ffa.2021.101953

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Pure Gauss sums and skew Hadamard difference sets2022

    • Author(s)
      Momihara Koji
    • Journal Title

      Finite Fields and Their Applications

      Volume: 77 Pages: 101932-101932

    • DOI

      10.1016/j.ffa.2021.101932

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] New constructions of strongly regular Cayley graphs on abelian non p-groups2021

    • Author(s)
      Momihara Koji
    • Journal Title

      Journal of Combinatorial Theory, Series A

      Volume: 184 Pages: 105514-105514

    • DOI

      10.1016/j.jcta.2021.105514

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Cameron-Liebler line classes with parameter x=\frac{(q+1)^2}{3}2021

    • Author(s)
      Feng Tao、Momihara Koji、Rodgers Morgan、Xiang Qing、Zou Hanlin
    • Journal Title

      Advances in Mathematics

      Volume: 385 Pages: 107780-107780

    • DOI

      10.1016/j.aim.2021.107780

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Powers of Gauss sums in quadratic fields2021

    • Author(s)
      Momihara Koji
    • Journal Title

      Journal of Number Theory

      Volume: - Pages: 331-352

    • DOI

      10.1016/j.jnt.2021.08.015

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] A new family of Hadamard matrices of order 4(2p^2+1)2021

    • Author(s)
      K. H. Leung, K. Momihara, Q. Xiang,
    • Journal Title

      Discrete Mathematics

      Volume: 344 Issue: 1 Pages: 112-163

    • DOI

      10.1016/j.disc.2020.112163

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A recursive construction for skew Hadamard difference sets2020

    • Author(s)
      K. Momihara
    • Journal Title

      The Electronic Journal of Combinatorics

      Volume: 27 Issue: 3

    • DOI

      10.37236/9058

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Strongly regular graphs from pseudocyclic association schemes2024

    • Author(s)
      籾原幸二
    • Organizer
      スペクトラルグラフ理論および周辺領域 第1 2 回研究集会
    • Related Report
      2023 Annual Research Report
  • [Presentation] Strongly regular graphs and association schemes from pseudocyclic association schemes2023

    • Author(s)
      Koji Momihara
    • Organizer
      2023 International Conference on Combinatorics and Finite Geometry
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The existence of Cameron-Liebler line classes with parameter x =(q+1)^2/32022

    • Author(s)
      籾原幸二
    • Organizer
      京都大学RIMS共同研究(研究集会)「有限群論,代数的組合せ論,頂点代数の研究」
    • Related Report
      2022 Research-status Report
  • [Presentation] Partial difference sets of Latin square type or negative Latin square type in abelian non p-groups2022

    • Author(s)
      Koji Momihara
    • Organizer
      The 6-th International Workshop on Algebraic Designs, Hadamard Matrices & Quanta
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 可換非p-群上のpartial difference setについて2021

    • Author(s)
      籾原幸二
    • Organizer
      第22回名古屋組合せ論セミナー(第9回 実験計画法とその周辺・オンラインセミナー)
    • Related Report
      2021 Research-status Report
    • Invited
  • [Remarks] Homepage of Koji Momihara

    • URL

      https://sites.google.com/view/kojimomihara/home

    • Related Report
      2023 Annual Research Report 2022 Research-status Report 2021 Research-status Report
  • [Remarks] Researchmap

    • URL

      https://researchmap.jp/7000016996

    • Related Report
      2022 Research-status Report
  • [Remarks] Homepage of Koji Momihara

    • URL

      https://www.educ.kumamoto-u.ac.jp/~momihara/

    • Related Report
      2020 Research-status Report

URL: 

Published: 2020-04-28   Modified: 2025-01-30  

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