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2000 Fiscal Year Final Research Report Summary

Error Correcting Codes

Research Project

Project/Area Number 11640123
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionYamaguchi University

Principal Investigator

KASHIWAGI Yoshimi  Fac.of Economics, Yamaguchi University Professor., 経済学部, 教授 (00152637)

Co-Investigator(Kenkyū-buntansha) SATO Yoshihisa  Fac.of Education, Associate Professor., 教育学部, 助教授 (90231349)
KIKUMASA Isao  Fac.of Science, Associate Professor., 理学部, 助教授 (70234200)
KATAYAMA Hirao  Fac of Science, Professor., 理学部, 教授 (00043860)
KITAMOTO Takuya  Fac.of Education, Associate Professor., 教育学部, 講師 (30241780)
KASAI Sinichi  Fac.of Education, Associate Professor., 教育学部, 助教授 (40224373)
Project Period (FY) 1999 – 2000
Keywordserror correcting code / indecomposability / bipartite graph / connectedness of graph / Boolean matrix / some relation
Research Abstract

One of our purposes of this research is to find an efficient method for determining the equivalence of codes. To handle this problem we were going to study three problems : (1) Calculation of weight distribution, (2) Characterization of code equivalence, (3) Characterization of indecomposability of code. We have obtained a result of the third problem. The result is as follows :
Let G be a standard generator matrix of a code C with dimension k and B=(b_<ij>) be a matrix next to the unitary matrix in G.Let b^^〜_<ij>=0∈Z if b_<ij>=0, let b^^〜_<ij>=1∈Z if b_<ij>=1 and let B^^〜=(b^^〜_<ij>). B^^〜 is a matrix over the ring Z of integers. B^^〜 can be regarded as the reduced adjacency matrix with the set V of row positions and the set W of column positions. Consider Γ_B=({V, W}, B^^〜) as a bipartite graph. Then we have that the indecomposability of C is equivalent to the connectedness of Γ_B. As an application of this result, we have that C is indecomposable if each row of B is not a zero vector and each component of some row of B is 1. We also have that C is indecomposable if each component of some column of B is 1. The following is an outline of the algorithm with which we can determine whether C is indecomposable or not :
1.Delete rows of B^^〜 all components of which are zero.
2.Go to 3 if B^^〜 does not have a row of weight 1. Else goto 1 with deleting all rows of weight 1.
3.In case that there exist different numbers t and u with b^^〜_<1t>=b^^〜_<1u>, then put b^^〜_<it>=b^^〜_<iu>=0 for i∈{1,2, ..., k}. Otherwise put b^^〜_<iu>=1 for i∈{1,2, ..., k}.
We also have the following result concerning Boolean matrices. If R is a Boolean matrix of length 4 which satisfies R^3=R^^-^t ∨ I and if n 【less than or equal】 4, then there exists a permutation σ with R^σ=Z, where Z is the upper triangular matrix.

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] 柏木芳美: "巡回符号の既約性"Proc.Summer Seminar on Lie Algebras…. 15. 23-35 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 柏木芳美: "R^3=R^^-^tVIを満たずブール行列"山口大学経済学雑誌掲載予定. 49. (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kasai,Sinichi: "On the microlocal structure of regular simple…"Tsukuba J.Math.. 24. 209-219 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kasai,Sinichi: "The b-function and the holonomy diagram of …"J.Algebra. 235. 1-14 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kitamoto,Takuya: "Efficient computation of the characteristic…"IEICE Trans.Fund.Elector.Com.Sci.. E82-A. 842-848 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kitamoto,Takuya: "The CHACM method for computing the…"IEICE Trans.Fund.Elector.Comm.Comp.. E83-A. 1405-1410 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 柏木芳美: "分野数学"自費出版. 193 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 柏木芳美: "基礎数学"自費出版. 320 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoshimi kashiwagi and Isao Kikumasa: "Indecomposebility of cyclic codes"Proc.Summer Seminar on Lie Algebras and related Topics. 15. 23-35 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshimi Kashiwagi: "Boolean matrices which satisfy R^3=R^^-^tVI"Yamaguchi J.Economics. 49(To appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sinichi Kasai: "On the microlocal structure of regular simple prehomogeneous vector space (GL(1)^2XSL(7), Δ_3+Δ^*_1)"Tsukuba J.Math.. 24. 209-219 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sinichi Kasai: "The b-function and the holonomy diagram of a regular simple prehomogeneous vector space (GLU)^2 X Spin (10), half-spin rep.+vector rep.)"J.Algebra. 235. 1-14 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takuya Kitamoto: "Efficient computation of the characteristic polynomial of a polynomial matrix"IEICE Trans.Fund.. E82-A. 842-848 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takuya Kitamoto: "The CHACM method for computing the characteristic polynomial of a polynomial matrix"IEICE Trans.Fund.. E83-A. 1405-1410 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshimi Kashimagi: "Bunyabetsu Sugaku"at my own expense. 193 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshimi Kashinagi: "Kiso Sugaku"at my own expense. 320 (2000)

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

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Published: 2002-03-26  

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