一类本原σ-LFSR 序列的构造与计数

有限域GF(2k)上本原σ-LFSR序列的分量序列均是二元域上具有相同极小多项式的m-序列,已知一条GF(2k)上本原σ-LFSR序列的距离向量,就可以用二元域上的m-序列构造它.研究了一类本原σ-LFSR序列——Z本原σ-LFSR序列距离向量的计算问题.给出了一种GF(2k)上n级Z本原σ-LFSR序列距离向量的计算方法,其主要思想是,利用GF(2k)上1级Z本原σ-LFSR序列的距离向量来计算n级Z本原σ-LFSR序列的距离向量.与其他现有方法相比,该方法的效率更高.更有价值的是,该方法也适用于GF(2k)上n级m-序列距离向量的计算.最后给出了GF(2k)上n级Z本原σ-LFSR序列的计数公式,说明其个数比GF(2k)上n级m-序列更多.

Construction and Enumeration of a Class of Primitive s-LFSR Sequences

The coordinate sequences of a primitive s-LFSR sequence over GF(2k) are m-sequences with the same minimal polynomial over GF(2),thus a primitive s-LFSR sequence over GF(2k) can be constructed by m-sequences over GF(2) if its interval vector is known.This paper studies the calculation of interval vectors of a class of primitive s-LFSR sequences—Z primitive s-LFSR sequences and presents an improved method to calculate the interval vectors of Z primitive s-LFSR sequences in order n over GF(2k),which uses the interval vectors of Z primitive s-LFSR sequences of order 1 to calculate that of Z primitive s-LFSR sequences in order n over GF(2k).In addition,it is more effective than other existing methods.More importantly,the new method can also be applied to the calculation of interval vectors of m-sequences over GF(2k).The enumeration formula of Z primitive s-LFSR sequences of order n over GF(2k) is also presented,which shows that the number of Z primitive s-LFSR sequences of order n is much larger than the number of m-sequences of order n over GF(2k).