| Project/Area Number |
11640123
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Yamaguchi University |
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Principal Investigator |
KASHIWAGI Yoshimi Fac.of Economics, Yamaguchi University Professor., 経済学部, 教授 (00152637)
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| 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)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
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| Keywords | error correcting code / indecomposability / bipartite graph / connectedness of graph / Boolean matrix / some relation |
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| 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.
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