2006 Fiscal Year Final Research Report Summary
Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras
| Project/Area Number |
15340002
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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 |
Algebra
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| Research Institution | Chiba University |
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Principal Investigator |
KITAZUME Masaaki Chiba University, Faculty of Science, Professor, 理学部, 教授 (60204898)
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| Co-Investigator(Kenkyū-buntansha) |
KOSHITANI Shigeo Chiba university, Faculty of Science, Professor, 理学部, 教授 (30125926)
NOZAWA Sohei Chiba university, Faculty of Science, Professor, 理学部, 教授 (20092083)
SUGIYAMA Ken-ichi Chiba university, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
HARADA Masaaki Yamagata University, Faculty of Mathematical Sciences, Associate Professor, 理学部, 助教授 (90292408)
CHIGIRA Naoki Muroran university of Engineering, Faculty of Engineering, Associate Professor, 工学部, 助教授 (40292073)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Finite Group / Simple Group / Sporadic Simple Group / Graph / Code / Lattice / Design / Vertex Operator Algebra |
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| Research Abstract |
We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :
1. We studied singly even codes of length 48 and odd unimodular lattices of rank 48 which have (resp. do not have) extremal neighbors. As a byproduct, we have constructed a new extremal code over Z/4Z.
2. We studied a putative extremal binary code of length 72. We showed that if there is a self-orthogonal 5-(72,16,78) design then the rows of its block-point incidence matrix generate an extremal doubly-even self-dual code of length 72.
3. We constructed new self-dual codes of length 100 invariant under the Hall-Janko group. We studied the binary code C(G, n) defined as the dual code of the code spanned by the sets of fixed points of involutions of a given group G. We showed that any G-invariant self-orthogonal code of length n is contained in C(G, n). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code and the above code for the Hall-Janko group, are obtained as C(G, n). Some new self-dual codes invariant under sporadic almost simple groups are constructed.
4. We constructed extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.
5. We studied maximum cocliques of sporadic rank 3 graphs and related designs. We gave some reconstructions of the Hall-Janko graph from the Witt system and the hexacode. Moreover we considered the maximum coclique design of the sporadic Suzuki graph. We constructed a new 3-(66,16,21) design with the automorphism group U_{3}(4):4, the unitary group over the 16-element field. By using this design, we gave a new construction of the sporadic Suzuki graph.
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Research Products
(9 results)
