The minimal polynomial of sequence obtained from componentwise linear transformation of linear recurring sequence

Let S = (s1, s2, . . . , sm, . . .) be a linear recurring sequence with terms in GF (q) and T be a linear transformation of GF (q) over GF (q). Denote T (S) = (T (s1), T (s2), . . . , T (sm), . . .). 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), 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) over GF (q). This bound is tight in some cases.