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

TL;DR

This paper corrects a previous result on the minimal polynomial of sequences transformed by linear maps over finite fields, providing new formulas and bounds for their linear complexity.

Contribution

It identifies errors in prior work, derives the minimal polynomial of transformed sequences given their original minimal polynomial, and establishes bounds on their linear complexity.

Findings

Counterexamples show previous results are incorrect in general.

Derived formulas for the minimal polynomial of T(S) based on the original sequence.

Provided an upper bound on the linear complexity of T(S).

Abstract

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.