Research On The Bernoulli Numbers Attached Formal Group

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(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.