Theory of algebraic curves motivated by coding theory
| Project/Area Number |
17540045
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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 |
Algebra
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| Research Institution | Kanagawa University |
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Principal Investigator |
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
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| Co-Investigator(Kenkyū-buntansha) |
KATO Takao Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10016157)
KOMEDA Jiryo Kanagawa Institute of Technology, Faculty of Engineering, Professor, 工学部, 教授 (90162065)
ISHII Naonori Nihon University, Faculty of Science and Engineering, Lecturer, 理工学部, 講師 (10339252)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | Algebraic geometry / Algebraic curve / Coding theory / Finite field / Positive characteristic / Hermitian curve / 正標数 / 国際研究者交流 / 韓国 |
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| Research Abstract |
Each result under this project is concerned with a Hermitian curve. Let q be an e th power of a prime number p, and F the finite field of q^2 elements. A Hermitian curve is a plane curve defined by the inhomogeneous equation y^q + y = x^<q+1> over F. The number of F-rational points of this curve is the maximum value with respect the genus, which is a reason why people prefer this curve in constructing example in coding theory, in finite geometry etc.
Previously we already determined the exact value of the minimum weight of any two-point code on the curve. Under this project, we tried to find exact value of the second Hamming weight of any two-point code on the Hermitian curve, and have succeeded we believe.
The other result is related with Galois theory for a separable morphism. For a point P in the ambient projective plane of the Hermitian curve, we consider the projection from the curve with center P. We proved that the projection forms a Galois covering if and only if the point is F-rational. Moreover if the F-rational point lies on the curve, then the Galois group is the direct sum of e copies of Z/pZ ; if F-rational point does not lie on the curve, then the Galois group is Z/(q+1).
When the point P is not F-rational, we consider the Galois closure of the projection. We have found out the Galois group for the field of the Galois closure over the field of the target line of the projection. If the point does not lie on the curve, the Galois group is the projective general linear group of an F-line ; if the point on the curve, it is the affine general linear group of an affine F-line.
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Report
(3 results)
Research Products
(9 results)
