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Research on families of maximal triply even codes and related mathematical structures

Research Project

Project/Area Number 17K05153
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionHirosaki University

Principal Investigator

Betsumiya Koichi  弘前大学, 理工学研究科, 准教授 (60364684)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords代数的組合せ論 / 符号理論 / 離散幾何 / 組合せデザイン / 散在型有限単純群 / グラフ理論 / 極大立方重偶符号 / 自己双対符号 / 散在型単純群 / 群論 / 代数的符号理論
Outline of Final Research Achievements

We obtained the following results.
(1) From some strongly regular graphs, we constructed triply even codes. We showed the maximality, the weight distribution, the automorphism groups and showed some other equipped structures. (2) On a series of triply even codes constructed from finite geometries, we showed some lemmas. (3) From triply even codes, we constructed some secret sharing schemes. (4) We made some sense of a lattice over a quadratic field constructed from a quaternary code in relations to triply even codes. (5) We calculated some subgroups of the Conway0 group which is the automorphism group of the Leech lattice.

Academic Significance and Societal Importance of the Research Achievements

これまでの研究を通して、立方重偶符号が共形場理論、有限群論、保型形式論、情報理論などの様々な数理構造と関連することが明らかとなっている。
立方重偶符号の研究を通して、頂点作用素代数の性質が明らかになることや、散在型単純群やLie型の有限単純群の系列に関する新たな視点が得られることは意義深い。

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (14 results)

All 2020 2019 2018 2017 Other

All Journal Article (3 results) Presentation (10 results) (of which Int'l Joint Research: 1 results) Remarks (1 results)

  • [Journal Article] tame 自己同型群について2020

    • Author(s)
      別宮耕一
    • Journal Title

      第31回有限群論草津セミナー報告集

      Volume: 31 Pages: 1-3

    • Related Report
      2019 Annual Research Report
  • [Journal Article] 拡張ハミング符号と正六百胞体との対応について2019

    • Author(s)
      別宮耕一
    • Journal Title

      第30回有限群論草津セミナー報告集

      Volume: 30 Pages: 37-42

    • Related Report
      2018 Research-status Report
  • [Journal Article] Triply even code の極大性判定2018

    • Author(s)
      別宮耕一
    • Journal Title

      第29回有限群論草津セミナー報告集

      Volume: - Pages: 14-17

    • Related Report
      2017 Research-status Report
  • [Presentation] tame 自己同型群について2019

    • Author(s)
      別宮耕一
    • Organizer
      第31回有限群論草津セミナー
    • Related Report
      2019 Annual Research Report
  • [Presentation] Maximal triply even code を構成する組合せ構造2019

    • Author(s)
      別宮耕一
    • Organizer
      Japanese Conference on Combinatorics and its Applications 2019
    • Related Report
      2019 Annual Research Report
  • [Presentation] H_4型コクセター群と3J_3の類似性2019

    • Author(s)
      別宮耕一
    • Organizer
      神戸学院大学群論特別セミナー
    • Related Report
      2019 Annual Research Report
  • [Presentation] On triply even codes2019

    • Author(s)
      別宮耕一
    • Organizer
      組合せ論的符号理論
    • Related Report
      2018 Research-status Report
  • [Presentation] 二次体の整数環上の格子について2019

    • Author(s)
      別宮耕一
    • Organizer
      千葉大 群論特別セミナー
    • Related Report
      2018 Research-status Report
  • [Presentation] A Distribution of known maximal triply even codes2019

    • Author(s)
      Koichi BETSUMIYA
    • Organizer
      Workshop on Finite Groups, vertex algebras and algebraic combinatorics
    • Related Report
      2018 Research-status Report
  • [Presentation] 拡張ハミング符号と正六百胞体との対応について2018

    • Author(s)
      別宮耕一
    • Organizer
      第30回有限群論草津セミナー
    • Related Report
      2018 Research-status Report
  • [Presentation] 正六百胞体格子と E_8 格子の対応について2018

    • Author(s)
      別宮耕一
    • Organizer
      福井表現論小研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] On some series of maximal triply even codes2018

    • Author(s)
      Koichi Betsumiya
    • Organizer
      Hadamard Matrices and Their Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Triply even code の極大性判定2017

    • Author(s)
      別宮耕一
    • Organizer
      第29回有限群論草津セミナー
    • Related Report
      2017 Research-status Report
  • [Remarks] DATABASE : Triply Even Codes of Length 48

    • URL

      http://www.st.hirosaki-u.ac.jp/~betsumi/triply-even/

    • Related Report
      2019 Annual Research Report 2018 Research-status Report 2017 Research-status Report

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Published: 2017-04-28   Modified: 2021-02-19  

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