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2003 Fiscal Year Final Research Report Summary

ALGEBRAIC CODING THEORY, COMBINATORIAL DESIGNS AND ASSOCIATION SCHEMES

Research Project

Project/Area Number 12440030
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTOHOKU UNIVERSITY (2003)
Kyushu University (2000-2002)

Principal Investigator

MUNEMASA Akihiro  TOHOKU UNIVERSITY, GRADUATE SCHOOL OF INFORMATION SCIENCES, PROFESSOR, 大学院・情報科学研究科, 教授 (50219862)

Co-Investigator(Kenkyū-buntansha) URAKAWA Hajime  TOHOKU UNIVERSITY, GRADUATE SCHOOL OF INFORMATION SCIENCES, PROFESSOR, 大学院・情報科学研究科, 教授 (50022679)
BANNAI Etsuko  KYUSHU UNIVERSITY, GRADUATE SCHOOL OF MATHEMATICS, ASSOCIATE PROFESSOR, 大学院・数理学研究院, 助教授 (00253394)
BANNAI Eiichi  KYUSHU UNIVERSITY, GRADUATE SCHOOL OF MATHEMATICS, PROFESSOR, 大学院・数理学研究院, 教授 (10011652)
HIRAKI Akira  OSAKA KYOIKU UNIVERSITY, FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (90294181)
HARADA Masaaki  YAMAGATA UNIVERSITY, FACULTY OF SCIENCE, ASSOCIATE PROFESSOR, 理学部, 助教授 (90292408)
Project Period (FY) 2000 – 2003
KeywordsALGEBRAIC COMBINATORICS / SELF-DUAL CODE / DISTANCE-REGULAR GRAPH / FINITE FIELD / SPIN MODEL / MODULAR FORM / LATTICE
Research Abstract

A generalization of the concept of a type II code has been obtained for arbitrary finite fields of characteristic two, and the mass formulas for such codes have been obtained. This allowed us to classify nearly all type II codes whose binary lengths are at most 32, and our results were presented in the 14th AAECC symposium in Australia. We have solved a long standing problem of the existence of tight spherical designs, and this result was presented in an international conference on combinatorics in the Netherlands in 2002.Our result uses the theory of modular forms, and we participated in an international conference on modular forms in 2004 in order to find more connections between combinatorics and the theory of modular forms. We also explored the connections between integral lattices with binary codes, and spherical designs with ordinary combinatorial designs. We have established a uniqueness of a certain self-orthogonal 5-design related to a binary self-dual code, and we have submitted a paper about this result.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] A.Munemasa: "Spherical 5-designs obtained from finite unitary groups"European Journal of Combinatorics. 25. 261-267 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] D.B.Dalan, M.Harada, A.Munemasa: "On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p"Journal of Combinatorial Designs. 11. 356-380 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] E.Bannai, M.Harada, A.Munemasa 他2名: "Type II codes over FZ+uF2 and applications to Hermitian modular forms"Abhandlungen Mathematischen Seminar Seminar der Universitat Hamburg. 73. 13-42 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Harada, A.Munemasa, K.Tanabe: "Extremal self-dual [40,20,8] codes with covering radius 7"Finite Fields and Their Applications. 10. 183-197 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Munemasa, M.Muzychuk, M.Klin, P.H.Zieschang: "Directed strongly regular graphs via coherent algebras"Linear Algebra and Applications. 377. 83-109 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] E.Bannai, H.Tanaka, O.Shimabukuro: "Finite analogues of non-Euclidean graphs and Ramanujan graphs"European Journal of Combinatorics. 25. 243-259 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Munemasa: "Spherical 5-designs obtained from finite unitary groups"European Journal of Combinatorics. 25. 261-267 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] D.B.Dalan, M.Harada, A.Munemasa: "On Hadamard matrices of order 2(p+1)with an automorphism of odd prime order p"Journal of Combinatorial Designs. 11. 356-380 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] E.Bannai, M.Harada, T.Ibukiyama, A.Munemasa, M.Oura: "Type II codes over F_2+uF_2 and applications to Hermitian modular forms"Abhandlungen Mathematischen Seminar der Universitat Hamburg. 73. 13-42 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Harada, A.Munemasa, K.Tanabe: "Extremal self-dual[40,20,8]codes with covering radius 7"Finite Fields and Their Applications. 10. 183-197 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Klin, A.Munemasa, M.Muzychuk, P.-H.Zieschang: "Directed strongly regular graphs via coherent algebras"Linear Algebra and Applications. 377. 83-109 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] E.Bannai, O.Shimabukuro, H.Tanaka: "Finite analogues of non-Euclidean graphs and Ramanujan graphs"European Journal of Combinatorics. 25. 243-259 (2004)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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