How to determine linear complexity and k-error linear complexity in some classes of linear recurring sequences

Several fast algorithms for the determination of the linear complexity of d-periodic sequences over a finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb F}_q$\end{document}, i.e. sequences with characteristic polynomial f(x) = xd − 1, have been proposed in the literature. In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic polynomial f(x) = (x − 1)d for an arbitrary positive integer d, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x) = (x^2+x+1)^{2^v}$\end{document} are presented. The result is then utilized to establish a fast algorithm for determining the k-error linear complexity of binary sequences with characteristic polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x^2+x+1)^{2^v}$\end{document}.