Asymptotic Behavior of Normalized Linear Complexity of Multi-sequences

Asymptotic behavior of the normalized linear complexity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{L_{\b{s}}(n)}{n}$\end{document} of a multi-sequence s̱ is studied in terms of its multidimensional continued fraction expansion, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\b{s}}(n)$\end{document} is the linear complexity of the length n prefix of s̱ and defined to be the length of the shortest multi-tuple linear feedback shift register which generates the length n prefix of s̱. A formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim \sup _{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$\end{document} together with a lower bound, and a formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim \inf_{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$\end{document} together with an upper bound are given. A necessary and sufficient condition for the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim_{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$\end{document} is also given.