关于循环差集的一个存在性定理

摘要

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F2d=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=2m.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)=x3和f(x)=x5,Im△f(x)是L*中一个循环差集.%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*.