正形置换的构造

Construction of orthomorphic permutations andA lower bound for number of them

给出了正形矩阵的若干性质,求出了n阶正形矩阵的有理标准形为diag{N1,N2,…,Ns},其中Ni是阶为ni的正形矩阵,(n1,n2,…,ns)为n的一个正递序分拆,且 sni=n;并利用对角正形矩阵的特点结合布尔函数构造了一批正形i=1置换,其中包括一类非线性正形置换.得到了2n阶正形置换的一个计数下界表达式为∏k (F2)2n22nk+2nk-1+nk+…+2n2+…+nk,其中n=2k时,ρ(n)={(2,2,…,｜Onii=1(n1,…,nk)∈ρ(n)2)};n=2k+1时,ρ(n)={(2,2,…,2,3),(2,2,…3,2),…,(3,2,…,2,2)}.

The several properties of orthomorphic matrices are given; the rational standard forms of orthomorphic matrices of order n are obtained, which is diag{N1,N2,...,Ns}, where Ni is an orthomorphic matrix of order  ni, (n1,n2,...,ns) is a positive increasing partition, and si=1?ni=n. A kind of orthomorphic permutations is constructed on the basis of the property of orthomorphic matrices and Boolean function, including a kind of nonlinear orthomorphic permutations. Furthermore, a lower bounds for the number of orthomorphic permutations with 2ndegree are obtained: (n1,...,nk)∈ρ(n)?∏ki=1?Oni(F2)2n22nk+2nk-1+nk+...+2n2+...+nk where ρ(n)={(2,2...,2)}; when n=2k, and ρ(n)={(2,2...,2,3),(2,2,...3,2),...,(3,2,...,2,2)} when n=2k+1.