关于循环差集的一个存在性定理

本文介绍了一个循环差集的存在性定理．主要结果是：设f(x)是域F2^d=L上一个置换多项式，如果f(x)是一个几乎完全非线性函数，则Im△f(x)是L^ =L＼{0}中一个循环差集当且仅当对任意a(≠0，1)∈Fq，|Sa|=q=2^m．这里，Sa={(x，y)|△f(x) a△f(y)=0}．△f(x)=f(x 1) f(x)|Sa|表示集合Sa的元素个数，作为应用，证明了在一定条件下，对f(x)=x^3。和f(x)=x^5，Im△f(x)是L^ 中一个循环差集．

An existence theorem on cyclic difference, sets

A new existence theorem on cyclic difference sets is introduced in this note. The main result is as follows: suppose that f(x) is permutation polynomial on the field F2d=L,and suppose that f(x)is an almost perfect nonlinear function, then Im△f(x) is a cyclic difference set in L*=L\{0} if and only if for each a(≠0,1)∈Fq,|Sa|=q, where Sa{(x,y):△f(x)+a△f(y)}=0},△f(x)=f(x)+f(x+ 1),and|Sa|denotes the cardinality of the set Sa. As applications, we proved that under certain conditions, when f(x)=x3 or f(x)=x5,Im△f(x) is a cyclic difference set in L*.