2003 Fiscal Year Final Research Report Summary
ALGEBRAIC CODING THEORY, COMBINATORIAL DESIGNS AND ASSOCIATION SCHEMES
| Project/Area Number |
12440030
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | TOHOKU UNIVERSITY (2003)
Kyushu University (2000-2002) |
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Principal Investigator |
MUNEMASA Akihiro TOHOKU UNIVERSITY, GRADUATE SCHOOL OF INFORMATION SCIENCES, PROFESSOR, 大学院・情報科学研究科, 教授 (50219862)
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| Co-Investigator(Kenkyū-buntansha) |
URAKAWA Hajime TOHOKU UNIVERSITY, GRADUATE SCHOOL OF INFORMATION SCIENCES, PROFESSOR, 大学院・情報科学研究科, 教授 (50022679)
BANNAI Etsuko KYUSHU UNIVERSITY, GRADUATE SCHOOL OF MATHEMATICS, ASSOCIATE PROFESSOR, 大学院・数理学研究院, 助教授 (00253394)
BANNAI Eiichi KYUSHU UNIVERSITY, GRADUATE SCHOOL OF MATHEMATICS, PROFESSOR, 大学院・数理学研究院, 教授 (10011652)
HIRAKI Akira OSAKA KYOIKU UNIVERSITY, FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (90294181)
HARADA Masaaki YAMAGATA UNIVERSITY, FACULTY OF SCIENCE, ASSOCIATE PROFESSOR, 理学部, 助教授 (90292408)
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| Project Period (FY) |
2000 – 2003
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| Keywords | ALGEBRAIC COMBINATORICS / SELF-DUAL CODE / DISTANCE-REGULAR GRAPH / FINITE FIELD / SPIN MODEL / MODULAR FORM / LATTICE |
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| Research Abstract |
A generalization of the concept of a type II code has been obtained for arbitrary finite fields of characteristic two, and the mass formulas for such codes have been obtained. This allowed us to classify nearly all type II codes whose binary lengths are at most 32, and our results were presented in the 14th AAECC symposium in Australia. We have solved a long standing problem of the existence of tight spherical designs, and this result was presented in an international conference on combinatorics in the Netherlands in 2002.Our result uses the theory of modular forms, and we participated in an international conference on modular forms in 2004 in order to find more connections between combinatorics and the theory of modular forms. We also explored the connections between integral lattices with binary codes, and spherical designs with ordinary combinatorial designs. We have established a uniqueness of a certain self-orthogonal 5-design related to a binary self-dual code, and we have submitted a paper about this result.
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