2010 Fiscal Year Final Research Report
Number theoretic and coding theoretic study of zeta functions appearing in applied mathematics
| Project/Area Number |
20540032
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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 | Kinki University |
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Principal Investigator |
CHINEN Koji Kinki University, 理工学部, 准教授 (30419486)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 線型符号 / 完全符号 / ゼータ関数 / 不変式環 / リーマン予想 |
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| Research Abstract |
This project dealt with a generalization of the theory of zeta functions for linear codes. We extended the consideration to all the polynomials which were invariant under the MacWilliams transform. Moreover we introduced a method to produce many invariant polynomials systematically from the existing codes which were not self-dual. Using this method, we considered the Riemann hypothesis for invariant polynomials which were obtained from some famous families of linear codes. They were the MDS codes, general Hamming codes and non-self-dual Golay codes. Some of them form a family "perfect codes". We could prove that, except for some sequences of the general Hamming codes, their invariant polynomials satisfied the Riemann hypothesis.
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