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2007 Fiscal Year Final Research Report Summary

Explicit construction of algebraic geometry codes

Research Project

Project/Area Number 18540038
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

UEHARA Tsuyoshi  Saga University, Faculty of Science and Engineering, Professor (80093970)

Co-Investigator(Kenkyū-buntansha) NAKAHARA Toru  Saga University, Faculty of Science and Engineering, Professor (50039278)
ICHIKAWA Takashi  Saga University, Faculty of Science and Engineering, Professor (20201923)
TERAI Naoki  Saga University, Faculty of Culture and Education, Associate Professor (90259862)
Project Period (FY) 2006 – 2007
KeywordsError-correcting codes / Algebraic geometry codes / Algebraic cnnstruntion of codes / Low density parity check codes / Power bases of integer rings / Siegel modular forms / Stanley-Reisner rings
Research Abstract

1. We have researched on Hermitian codes which are constructed by Hermitian curves, and have found a new method of constructing of them. By our method we have constructed new Hermitian codes different from known ones, that is, one of the one-point type, and we have further computed a lower bound for their minimum distances. As a result, we have proved that our Hermitian codes have better properties than ones of the one-point type.
2. We carried out an investigation on finding an explicit linear basis of the trace-norm code, and have found its explicit form when the number of variables is less than or equal to 3.
3. We have researched on algebraic construction of low density parity check codes, and have shown methods of constructions of them by vector spaces over a finite field and by non-abelian groups.
4. We have studied on structure of integral bases of algebraic number fields, and have proved that there is only one case that the integer ring of a 2-elementary abelian field of degree greater than or equal to 8 is generated by a single integer.
5. We have researched on Siegel modular forms, and have described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6.
6. We have studied Stanley-Reisner rings which are locally complete intersections. As a result, we proved that locally complete intersection Stanley-Reisner rings are complete intersections if the corresponding simplicial complexes are of dimension more than 1 and are connected.

  • Research Products

    (13 results)

All 2007 2006

All Journal Article (8 results) (of which Peer Reviewed: 3 results) Presentation (5 results)

  • [Journal Article] Construction of evaluation codes from Hermitian curves2007

    • Author(s)
      K.-H. Park, Tsuyoshi Uehara
    • Journal Title

      Kyushu Journal of Mathematics 56

      Pages: 415-429

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Construction of evaluation codes from Hermitian curves2007

    • Author(s)
      Kyoung Ho, Park, Tsuyoshi, Uehara
    • Journal Title

      Kyushu J. Math vol.56, no.2

      Pages: 415-429

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A family of Schottky groups arising from the hypergeometric equatio2006

    • Author(s)
      Takashi Ichikawa, M. Yoshida
    • Journal Title

      Proceedings of American Mathematical Society 134

      Pages: 2271-2280

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Stanley-Reisner rings with large multiplicities are Cohen-Macaulay2006

    • Author(s)
      Naoki Terai, K. Yoshida
    • Journal Title

      Journal of Algebra 301

      Pages: 493-508

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] On power integral bases of the 2-elementary abelian extension fields2006

    • Author(s)
      Yasuo, Motoda, Toru, Nakahara, Kyoung Ho, Park
    • Journal Title

      Trends in Mathematics 9-1

      Pages: 55-63

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A family of Schottky groups arising from the hypergeometric equation2006

    • Author(s)
      Takashi, Ichikawa, Masaaki, Yoshida
    • Journal Title

      Proc. Amer. Math. Soc Soc. 134

      Pages: 2271-2280

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Stanley-Reisner rings and Cohen-Macaulay covers2006

    • Author(s)
      Naoki, Terai, Ken-ichi, Yoshida
    • Journal Title

      Communication in Algebra 34, no.7

      Pages: 2673-2681

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Buchsbaum Stanley-Reisner rings with minimal multiplicity2006

    • Author(s)
      Naoki, Terai, Ken-ichi, Yoshida
    • Journal Title

      Proc. Amer. Math. Soc Soc. 134, no.1

      Pages: 55-65

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Congruence and p-adic properties of Siegel modular form2007

    • Author(s)
      Takashi, Ichikawa
    • Organizer
      Seminar on number theoretic geometry
    • Place of Presentation
      Hokkaido Univ
    • Year and Date
      20071221-21
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] ハッセの問題について2007

    • Author(s)
      元田 康夫、中原 徹、上原 健、シャー・サイエッド・イナヤット・アリ
    • Organizer
      研究集会「代数的整数論とその周辺」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-12-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On Hasse's Problem2007

    • Author(s)
      Y., Motoda, T., Nakahara, T., Uehara, S.I.A., Shah
    • Organizer
      Symposium on Algebraic Number Theory and its Related Area
    • Place of Presentation
      Kyoto Univ
    • Year and Date
      2007-12-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Orbits of irrationals by the modular group in a real quadratic field2007

    • Author(s)
      Toru, Nakahara
    • Organizer
      The 17th International Conference of Jangjeon Mathematical Society
    • Place of Presentation
      Kyungpook National University the Republic of Korea
    • Year and Date
      2007-07-08
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Lower bound theorem and regularity of monomial ideals2007

    • Author(s)
      Naoki, Terai
    • Organizer
      Workshop "Castelnuovo-Mumford regularity and applications"
    • Place of Presentation
      Leipzig, Germany
    • Year and Date
      2007-06-16
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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