设f(x)=x^n+c_(n-1)x^(n-1)+…+C_0是Z/(2~e)上首一多项式,适合关系式a_(i+n)=-(c_0a_i+c_1a_(i+1)+…+c_(n-1)a_(i+n-1)),i=0,1,2,…(1)的Z/(2~e)上序列a=(a_0,a_1,…)称由f(x)生成的线性递归序列,由f(x)生成的Z/(2~e)上的所有序列的集合记为G(f(x))_e,并记G’(f(x))_e={a∈G(f(x))_e│a≠0 mod 2}.递归式(1)等价于关系式f(x)a=0=(0,0,…),其中x表示移位算子,即xa=(a_1,a_2,a_3,…).Z/(2~e)上序列a有唯一权位分解a=a_0+a_12+…+a_(e-1)2^(e-1),其中a_i=(a_(i0),a_(i1),…)是0,1序列,并称a_i是a的第i权位序列,称a_(e-1)为a的最高权位序列.对Z/(2~e)上首一n次多项式f(x),若f(0)(即c_0)是可逆元,则由文献[1],f(x)的周期per(f(x))_e≤2^(e-1)(2~n-1).当per(f(x))=2^(e-1)(2~n-1)时,称f(x)是Z/(2~e)上n次本原多项式,并称G’(f(x))_e中序列为f(x)生成的本原序列.
In this paper,we study the ideal of the characteristic polynomials and the period of the linear recurring sequences over Z/(m).Our tool is ring theory and we obtain the structure of the ideal of the characteristic poly- nomials and the period of linear recurring sequences.
