2006 Fiscal Year Final Research Report Summary
Research On The Bernoulli Numbers Attached Formal Group
| Project/Area Number |
15540019
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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 | Nagoya University |
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Principal Investigator |
SATOH Junya Nagoya University, Grad.Sch.Info., Associate Professor, 大学院情報科学研究科, 助教授 (20235352)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Hiroyuki Nagoya University, Grad.Sch.Info., Professor, 大学院情報科学研究科, 教授 (00190538)
MATSUBARA Yo Nagoya University, Grad.Sch.Info., Professor, 大学院情報科学研究科, 教授 (30242788)
YOSHINOBU Yasuo Nagoya University, Grad.Sch.Info., Associate Professor, 大学院情報科学研究科, 助教授 (90281063)
MATSUMOTO Kohji Nagoya University, Grad. Sch.Math., Professor, 大学院多元数理科学研究科, 教授 (60192754)
TANIGAWA Yoshio Nagoya University, Grad. Sch.Math., Assciate Professor, 大学院多元数理科学研究科, 助教授 (50109261)
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| Project Period (FY) |
2003 – 2006
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| Keywords | formal group / number theory / zeta function / Bernoulli numbers / Kronecker density / power residue / reciprocity law / code theory |
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| Research Abstract |
(1) We extend a well-known distribution relation for ordinary Bernoulli polynomials to that of Bernoulli polynomials attached to formal group.
(2) A conflict-avoiding code (CAC) C of length n with weight k is a family of binary sequences of length n and weight k satisfying P_0 t_<njl> x_<it>x_<j ; t+s>・, for any distinct codewords X_i= (X_<iO>, X_<il>;:::; X_<i ; nj1>) and X_j = (x_<jo>, x_<jl>;:::; X_<j ; nj1>) in C and for any integer s, where the subscripts are taken modulo n. A CAC with maximal code size for given n and k is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the case, = 1 is treated, and various direct and recursive constructions of optimal CACs for weight k = 4 and 5 are obtained by providing constructions of CACs for general weight k. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.
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