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Design of BCH-Goppa Decoding Algorithms in Terms of the Tau-functions over Finite Fields

Research Project

Project/Area Number 09559011
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section展開研究
Research Field 広領域
Research InstitutionOsaka University

Principal Investigator

NAKAMURA Yoshimasa  Graduate School of Engineering Science, Osaka University Professor, 大学院・基礎工学研究科, 教授 (50172458)

Co-Investigator(Kenkyū-buntansha) HIROTA Ryogo  Faculty of Engineering and Science, Waseda University, Professor, 理工学部, 教授 (00066599)
IMAI Jun  Communication Science Laboratories, NTT Corporation, Chief Researcher, コミュニケーション科学研究所, 主任研究員
SHIROTA Norihisa  Media Processing Institute, Sony Corporation, General Manager and Chief Research Scientist, メディアプロセシング研究所, 統括部長主幹研究員
OKAZAKI Ryutaro  Faculty of Engineering, Doshisha University, Lecturer, 工学部, 専任講師 (20268113)
高橋 大輔  龍谷大学, 理工学部, 助教授 (50188025)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥6,900,000 (Direct Cost: ¥6,900,000)
Fiscal Year 1999: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1997: ¥3,000,000 (Direct Cost: ¥3,000,000)
KeywordsDiscrete-Time Integrable Systems / Algorithms / BCH-Goppa Decoding / Continued Fraction Expansion / Orthogonal Polynomials / 連分散展開 / タウ関数 / BCH-Goppa複号法 / 戸田分子方程式 / QDアルゴリズム
Research Abstract

In 1997 the following results are given. To decrease the computational complexity of the algorithm based on moment problems introduced by the head of investigator of this project a new decoding algorithm is formulated by using the finite Toda molecule equation over finite fields, namely, the quotient difference algorithm over finite fields It is checked that the algorithm can be successfully applied to various examples including the Goppa code. Secondly, the dynamics of the finite Toda molecule equation over finite fields is considered. There exist conserved quantities and periodic orbits where each period is equal to the order of finite field or its divisor. The periods and the correspondence to the BCH-Goppa decoding are completely classified by the conserved quantities. Moreover, a relationship is discussed between the first component of the eigenvector of the Lax pair of the finite Toda molecule and the value of error appearing the BCH-Goppa code.
In 1998, the results obtained in 19 … More 97 are published as a Phys. Lett. A paper entitled "Dynamics of the finite Toda molecule over finite fields and a decoding algorithm". It is shown that the modified KdV (mKdV) equation induces a continued fraction expansion which is different from that given by the Toda molecule. The mKdV equation describes a one-parameter deformation of the orthogonal polynomials named the symmetric Szego polynomials. An integrable discretization of the mKdV equation is found.
In 1999, an integrable discretization of the Schur flow is given which is a generalization of the discrete mKdV equation and is corresponding to the general Szego polynomials. Hirota's bilinear form and the tau-function plays the central role. With the help of the discrete Schur flow a new O(N^2) algorithm for computing continued fraction expansions of given Herglotz class functions is designed Moreover, a new algorithm for solving algebraic equations is formulated by the discrete mKdV equation, which will be published in Inverse Problems. Less

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report

Research Products

(37 results)

All Other

All Publications (37 results)

  • [Publications] Y. Nakamura: "Calculating Laplace transforms in terms of the Toda moleclue"SIAM J. Sci. Comput.. 20巻. 306-317 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Mukaihara and Y. Nakamura: "Integrable discretization oh the modified KdV equation and applications"Inverse Problem(to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura: "Algorithms associated with arithmetic, geometric, and harmonic means and integrable systems"J. Comput. Appl. Math. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 中村佳正: "アルゴリズムと可積分系"システム制御情報学会誌. 43巻10号. 584-592 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura and L. Faybusovich: "On explicitly solvable gradient systems of Moser-Karmarkar type"J. Match. Anal. Appl.. 205. 88-106 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura: "Jacobi algorithm for symmetric eigenvalue problem and integrable gradient system of Lax form"Japan J. Indust. Appl. Math.. 14. 159-168 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura: "Integrable deformation of Gaussian distribution and the Ornstein-Uhlenbech process"Lett. Math. Phys.. 43. 1-5 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura, K. Kajiwara, and H. Shiotani: "On an integrable discretization of the Rayleigh quotient gradient system and the power method with a shift"J. Comput. Appl. Math.. 96. 77-90 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Kondo, Y. Nakamura: "An extension of Steffensen's iteration and its cpmputational complexity"Interdiscip. Inform. Sci.. 4. 129-138 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura, A. Mukaihira: "Dynamics of the finite Toda molecule over finite fields and a decoding algorithm"Phys. Lett. A. 249. 295-302 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Minesaki, Y. Nakamura: "On integrable discretization of integrable systems with separatrix"Phys. Lett. A. 251. 300-310 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura: "Calculating Laplace transforms in terms of the Toda molecule"SIAM J. Sci. Comput.. 20. 306-317 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Mukaihira and Y. Nakamura: "Integrable discretization of the modified KdV equation and applications"Inverse Problem. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakamura: "Algorithms associated with arithmetic, geometric and harmonic means and integrable systems"J. Comput. Appl. Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J. Imai: "On cohomological weight estimation methods for moduli spaces for codes"Proceedings of the 19th Symposium on Information Theory and its Spplications. 461-464 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J. Imai: "The generalization of coding theory and group representations"Proceedings of the 17th Symposium on Information Theory and its Applications. 1-4 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Hirota, M. Iwao et al.: "From integrability to chaos in a Lotka-Volterra cellular automaton"Phys. Lett. A. 226. 39-44 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Hirota: "Conserved quantities of "random-time" Toda equation"J. Phys. Soc. Japan. 66. 283-284 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Hirota: ""Molecule solution"of coupled modified KdV equaton"J. Phys. Soc. Japan. 66. 2530-2532 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Louboutin and R. Okazaki: "The class number one problem for some non-abelian normal CM-fields of $2$-power degrees"Proc. London Math. Soc.. 76. 523-548 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Okazaki: "On a Shintani decomposition for a cubic field defined by $X^3+kX-1=0$"Number Theory-Diophantine, Computational and Algebraic Aspects, K. ~g\、"ory et al. eds. de Gruyter. 445-451 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Nakamura: "Calculating Laplace transforms in terms of the Toda molecule"SIAMJ,Sci.Comput.. 20巻. 306-317 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] A.Mukaihira and Y.Nakamura: "Integrable discretization of the modified KdV equation and applications"Inverse Problem. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Nakamura: "Algorithms associated with arithmetic,geometric and hamonic means and integrable systems"J.Comput.Appl.Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] 中村佳正: "アルゴリズムと可積分系"システム制御情報学会誌. 43巻・10号. 584-592 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y. Nakamura: "Integrable deformation of Gaussian distribution and the Ornstein-Uhlenbeck process" Letters in Mathematical Physics. 43. 1-5 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y. Nakamura, K. Kajiwara, H. Shiotani: "On an integrable discretization of the Rayleigh quotient gradient system and the power method with a shift" Journal of Computational and Applied Mathematics. 96. 77-90 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K. Kondo, Y. Nakamura: "An extension of Steffensen's iteration and its computational complexity" Interdisciplinary Information Sciences. 4. 129-138 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y. Nakamura, A. Mukaihira: "Dynamics of the finite Toda molecule over finite fields and a decoding algorithm" Physics Letters A. 249. 295-302 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y. Minesaki, Y. Nakamura: "On integrable discretization of integrable systems with separatorix" Physics Letters A. 250. 300-310 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Nakamura, L.Faybusovich: "On explicitly solvable gradient systems of Moser-Karmlarkar type" Journal of Mathematical Analysis and Applications. Vol.205. 88-106 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Nakamura: "Jacobi algorithm for symmetric eigenvalue problem and integrable gradient system of Lax form" Japan Journal of Industrial and Applied Mathematics. Vol.14. 159-168 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Nakamura: "Calculating Laplace transforms in terms of the Toda molecule" SLAM Journal on Scientific Computing. 発表予定.

    • Related Report
      1997 Annual Research Report
  • [Publications] 中村 佳正: "ソリトン理論と数値計算法" 電子情報通信学会誌. 80巻11号. 1143-1146 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] R.Hirota, M.Iwao他: "From integrability to chaos in a Lotka-Volterr a cellular automaton" Physirr Letters A. Vol.226. 39-44 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] J.Imai: "Decording methods derived by integrability" Proceedings of The 20th Symposium on Information Theory and Its Applications. 157-160 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 広田 良吾: "差分学入門 情報化時代の微積分学" 培風館, 229 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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