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Efficient Decoding Method of Some Algebraic Geometry Codes

Research Project

Project/Area Number 02650262
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 情報工学
Research InstitutionToyohashi University of Technology

Principal Investigator

SAKATA Shojiro  Toyohashi University of Technology, Department of Knowledge-Based Information Engineering, Professor, 工学部, 教授 (20064157)

Project Period (FY) 1990 – 1991
Project Status Completed (Fiscal Year 1991)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1991: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1990: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsError-Correcting Codes / Algebraic Geometry Codes / Algebraic Curves / Computational Complexity of Decoding Algorithm / Fast Decoding Algorithm / Two-Dimensional Berlekamp-Massey Algorithm / 2D Syndrome Array / Total Ordering over the 2D Integral Lattice / 2次元BerlekampーMasseyアリゴリズム
Research Abstract

Recently a sequence of error-correcting codes that have good error-correction performance and high coding rate have been discovered using algebraic geometry and ideas of a russian coding theorist V. D. Goppa. These codes are a class of algebraic codes which are derived from algebraic curves over finite fields and are called algebraic geometry codes. Several investigators in Europe as well as in Japna have proposed decoding methods for some algebraic geometry codes quite recently. The computational complexties in most of these decoding methods are of the order of n^3 or more, where n is the code length. On the other hand, some investigators gave a fast algorithm by applying our 2D Berlekamp-Massey algorithm. The 2D Berlekamp-Massey algorithm which we proposed before solves th e 2D version of the problem treated by the (1D) Berlekamp-Massey algorithm, that is synthesis of 2D linear feedback shift register capable of generating a given finite 2D array. Both these algorithms in principle have the same computational complexity of the order of n^2, where n is the size of the given 1D or 2D array in this case. In the progression of this research we have revealed some more applications of our algorithm to decode several other algebraic geometry codes by making clear some novel aspects of the algorithm and by giving some new versions of it. We have shown that our methods can be applied to decode efficiently several new algebraic geometry codes which just have been discovered by other coding theorists. The computational complexities of our decoding methods are less than those of the previous methods. Furthermore, we have given several extensions of our algorithm to apply them to decode some other codes which have not been constructed yet but have the possibility of being invented.

Report

(3 results)
  • 1991 Annual Research Report   Final Research Report Summary
  • 1990 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] Shojiro Sakata: "Decoding Binary 2D Cyclic Codes by the 2D Berlekamp-Massey Algorithm" IEEE Transactions on Information Theory. 37. 1200-1203 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Two-Dimensional Shift Register Synthesis and Groebner Bases for Polynomial Ideals over an Interger Residue Ring" Discrete Applied Mathematics. 33. 191-203 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "A Groebner Basis and a Minimal Polynomial Set of a Finite nD Array" Applied Algebra,Algebraic Algorithms and Error-Correcting Codes,Proc.AAECC-8 (Ed.S.Satata),Springer Verlag. 280-291 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "On Application of the 2D BerlekampーMassey Algorithm to Decoding Some Codes Defined on Algebraic Curves" 第13回情報理論とその応用シンポジウム予稿集. 469-472 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Finding a Minimal Polynomial Vector Set of a Vector of nD Arrays" Applied Algebra,Algebraic Algorithms and Error-Correcting Codes,Proc.AAECC-9 (Eds.H.F.Mattson,T.Mora,T.R.N.Rao),Springer Verlag. 414-425 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "2D Linear Recurrences Restricted to a Sector of the Quadrant" 第14回情報理論とその応用シンポジウム予稿集. 321-324 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "A Fast Version of the Modified Skorobogatov Vladut Algorithm to Decode Some Codes from Algebraic Curves" IEEE Transactions on Information Theory.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Decoding Binary 2D Cyclic Codes by the 2D Berlekamp-Massey Algorithm." IEEE Transactions on Information Theory. 37, No. 4. 1200-1203 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Two-Dimensional Shift Register Synthesis and Groebner Bases for Polynomial Ideals over an Integer Residue Ring." Discrete Applied Mathematics. 33. 191-203 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "A Groebner Basis and a Minimal Polynomial Set of a Finite nD Array. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes" 8th International Conference, AAECC-8, 1990, Proceedings. Lecture Notes in Computer Science, Vol. 508. 280-291 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "On Application of the 2D Berlekamp-Massey Algorithm to Decoding Some Codes Defined on Algebraic Curves." Proceedings of 13th Symposium of Information and Its Appllications. 469-472 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Finding a Minimal Polynomial Vector Set of a Vector of nD Arrays. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes" 9th International Conference, AAECC-9, 1991, Proceedings. Lecture Notes in Computer Science, Vol. 539. 414-425 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "2D Linear Recurrences Restricted to a Sector of the Quadrant" Proceedings of 14th Symposium of Information and Its Applications. 321-324 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "A Fast Version of the Modified Skorobogatov Vladut Algorithm to Decode Some Codes from Algebraic Curves." IEEE Transactions on Information Theory.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Shojiro Sakata: "Decoding Binary 2D Cyclic Codes by the 2D BerlekampーMassey Algorithm" IEEE Transactions on Information Theory. 37. 1200-1203 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Shojiro Sakata: "TwoーDimensional Shift Register Synthesis and Groebner Bases for Polynomial Ideals over an Integer Residue Ring" Discrete Applied Mathematics. 33. 191-203 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] S.Sakata(ed.),Shojiro Sakata: "A Groebner Basis and a Minimal Polynomial Set of a Finite nD Array" Applied Algebra,Algebraic Algorithms and ErrorーCorrecting Codes,Proc.AAECCー8,Springer Verlag. 280-291 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Shojiro Sakata: "On Application of the 2D BerlekampーMassey Algorithm to Decoding Some Codes Defined on Algebraic Curves" 第13回情報理論とその応用シンポジウム予稿集. 469-472 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] H.F.Mattson,T.Mora,T.R.,N.Rao(ed.)Shojiro Sakata: "Finding a Minimal Polynomial Vector Set of a Vector of nD Arrays" Applied Algebra,Algebraic Algorithms and ErrorーCorrecting Codes,Proc.AAECCー9,Springer Verlag. 414-425 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Shojiro Sakata: "2D Linear Recurrences Restricted to a Sector of the Quadrant" 第14回情報理論とその応用シンポジウム予稿集. 321-324 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Shojiro Sakata: "A Fast Version of the Modified Skorobogatov Vladut Algorithm to Decode Some Codes from Algebraic Curves" IEEE Transactions on Information Theory.

    • Related Report
      1991 Annual Research Report
  • [Publications] Shojiro Sakata: "Decoding Binary 2D Cyclic Codes by the 2D BerlekampーMassey Algorithm" IEEE Transactions on Information Theory. (1991)

    • Related Report
      1990 Annual Research Report
  • [Publications] Shojiro Sakata: "TwoーDimensional Shift Register Synthesis and Groebner Bases for Polynomial Ideals over an Integer Residue Ring" Proceedings of AAECCー7.

    • Related Report
      1990 Annual Research Report
  • [Publications] Shojiro Sakata: "A Groebner Basis and a Minimal Polynomial Set of a Finite nD Array" Proceedings of AAECCー8.

    • Related Report
      1990 Annual Research Report
  • [Publications] Shojiro Sakata: "Decoding Some AlgebraicーGeometric Codes by the TwoーDimensional BerlekampーMassey Algorithm" Proceedings of the 13th Symposium on Imformation Theory and Its Applications. 469-472 (1991)

    • Related Report
      1990 Annual Research Report

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Published: 1990-04-01   Modified: 2016-04-21  

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