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Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras

Research Project

Project/Area Number 15340002
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

KITAZUME Masaaki  Chiba University, Faculty of Science, Professor, 理学部, 教授 (60204898)

Co-Investigator(Kenkyū-buntansha) KOSHITANI Shigeo  Chiba university, Faculty of Science, Professor, 理学部, 教授 (30125926)
NOZAWA Sohei  Chiba university, Faculty of Science, Professor, 理学部, 教授 (20092083)
SUGIYAMA Ken-ichi  Chiba university, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
HARADA Masaaki  Yamagata University, Faculty of Mathematical Sciences, Associate Professor, 理学部, 助教授 (90292408)
CHIGIRA Naoki  Muroran university of Engineering, Faculty of Engineering, Associate Professor, 工学部, 助教授 (40292073)
山田 裕理  一橋大学, 経済学研究科, 教授 (50134888)
宗政 昭弘  東北大学, 情報科学研究科, 教授 (50219862)
Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥9,400,000 (Direct Cost: ¥9,400,000)
Fiscal Year 2006: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2004: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2003: ¥2,300,000 (Direct Cost: ¥2,300,000)
KeywordsFinite Group / Simple Group / Sporadic Simple Group / Graph / Code / Lattice / Design / Vertex Operator Algebra / モンスター単純群
Research Abstract

We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :
1. We studied singly even codes of length 48 and odd unimodular lattices of rank 48 which have (resp. do not have) extremal neighbors. As a byproduct, we have constructed a new extremal code over Z/4Z.
2. We studied a putative extremal binary code of length 72. We showed that if there is a self-orthogonal 5-(72,16,78) design then the rows of its block-point incidence matrix generate an extremal doubly-even self-dual code of length 72.
3. We constructed new self-dual codes of length 100 invariant under the Hall-Janko group. We studied the binary code C(G, n) defined as the dual code of the code spanned by the sets of fixed points of involutions of a given group G. We showed that any G-invariant self-orthogonal code of length n is contained in C(G, n). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code and the above code for the Hall-Janko group, are obtained as C(G, n). Some new self-dual codes invariant under sporadic almost simple groups are constructed.
4. We constructed extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.
5. We studied maximum cocliques of sporadic rank 3 graphs and related designs. We gave some reconstructions of the Hall-Janko graph from the Witt system and the hexacode. Moreover we considered the maximum coclique design of the sporadic Suzuki graph. We constructed a new 3-(66,16,21) design with the automorphism group U_{3}(4):4, the unitary group over the 16-element field. By using this design, we gave a new construction of the sporadic Suzuki graph.

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (14 results)

All 2007 2005 2004 2003 Other

All Journal Article (10 results) Publications (4 results)

  • [Journal Article] Extremal Self-Dual Codes of Length 64 through Neighbors and Covering Radii2007

    • Author(s)
      N.Chigira, M.Harada, M.hitazume
    • Journal Title

      Designs, Codes and Cryptography 42

      Pages: 93-101

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Finite permutation groups and self-orthogonal codes2007

    • Author(s)
      N.Chigira, M.Harada, M.Kitazume
    • Journal Title

      Journal of Algebra 309

      Pages: 610-621

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Extremal Self-Dual Codes of Length 64 through Neighbors and Covering Radii2007

    • Author(s)
      N.Chigira, M.Harada, M.Kitazume
    • Journal Title

      Designs, Codes and Cryptography 42

      Pages: 93-101

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On some self-dual codes and unimodular lattices in dimension 482005

    • Author(s)
      M.Harada, M.Kitazume, A.Munemasa, B.Venkov
    • Journal Title

      European Journal of Combinatorics 26

      Pages: 543-557

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] On some self-dual codes and unimodular lattices in dimension 482005

    • Author(s)
      M.Harada, M.Kitazume, A.Munemasa, R.Venkov
    • Journal Title

      European Journal of Combinatorics 26

      Pages: 543-557

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a 5-design related to an extremal doubly even self-dual code of length 722004

    • Author(s)
      M.Harada, M.Kitazume, A.Munemasa
    • Journal Title

      Journal of Combinatorial Theory (A) 107

      Pages: 143-146

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Classification of Type II Z_6-Codes of Length 82004

    • Author(s)
      M.Kitazume, T.Ooi
    • Journal Title

      AKCE International Journal of Graphs and Combinatorics 1

      Pages: 35-40

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] On a 5-design related to an extremal doubly even self-dualcode of length 722004

    • Author(s)
      M.Harada, M.Kitazume, A.Munemasa
    • Journal Title

      Journal of Combinatorial Theory (A) 107

      Pages: 143-146

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 3-state Potts model, Moonshine vertex operator algebra and 3A-elements of the Monster group2003

    • Author(s)
      N.Kitazume, C.H.Lam, H.Yamada
    • Journal Title

      International Mathematical Research Notices 23

      Pages: 1269-1303

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 3-state Potts model, Moonshine vertex operator algebra and 3A-elements of the Monster group2003

    • Author(s)
      M.Kitazume, C.H.Lam, H.Yamada
    • Journal Title

      International Mathematical Research Notices 23

      Pages: 1269-1303

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Publications] M.Kitazume, C.H.Lam, H.Yamada: "3-state Potts model, Moonshine vertex operator algebra and 3A-elements of the Monster group"International Mathematical Research Notices. 23. 1269-1303 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] C.H.Lam, H.Yamada: "Z_3-twisted representations of lattice vertex operator algebras"Communications in Algebra. 31. 3001-3020 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] D.B.Dalan, M.Hadara, A.Munemasa: "On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p"Journal of Combinatorial Designs. 11. 367-380 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Koshitani: "Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups"Communications in Algebra. 31. 2229-2243 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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