线性递归序列的分量线性变换求序列的最小多项式
The minimal polynomial of sequence obtained from componentwise linear transformation of linear recurring sequence
摘要：
设$S=(s_1,s_2,...,s_m,...)$是一个线性递归序列，其项在$GF(q^n)$中，$T$是$GF(q^n)$在$GF(q)$上的线性变换。表示为$T(S)=(T(s_1),T(s_2),...,T(s_m),...)$。本文首先给出反例来证明[A. M. Youssef和G. Gong，On linear complex of sequences over $GF(2^n)$，Theoretical计算机科学，352（2006），288-292]一般是不正确的，因为该论文中的引理3是不正确的。然后，在已知无重根的$S$的极小多项式的典型因式分解的条件下，确定了$T(S)$的极小多项式，从而给出了上述文献中主要考虑但未正确解决的问题的解决方法.另外，作为特殊情况，如果$S$的极小多项式是本原的，我们确定了$T(S)$的极小多项式.最后，当$T$穷尽了$GF(q^n)$在$GF(q)$上的所有可能的线性变换时，给出了$T(S)$的线性复杂度的上界.这个界限在某些情况下是紧的。
英文摘要：
Let $S=(s_1,s_2,...,s_m,...)$ be a linear recurring sequence with terms in $GF(q^n)$ and $T$ be a linear transformation of $GF(q^n)$ over $GF(q)$. Denote $T(S)=(T(s_1),T(s_2),...,T(s_m),...)$. In this paper, we first present counter examples to show the main result in [A.M. Youssef and G. Gong, On linear complexity of sequences over $GF(2^n)$, Theoretical Computer Science, 352(2006), 288-292] is not correct in general since Lemma 3 in that paper is incorrect. Then, we determine the minimal polynomial of $T(S)$ if the canonical factorization of the minimal polynomial of $S$ without multiple roots is known and thus present the solution to the problem which was mainly considered in the above paper but incorrectly solved. Additionally, as a special case, we determine the minimal polynomial of $T(S)$ if the minimal polynomial of $S$ is primitive. Finally, we give an upper bound on the linear complexity of $T(S)$ when $T$ exhausts all possible linear transformations of $GF(q^n)$ over $GF(q)$. This bound is tight in some cases.
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