Asymptotic Behavior of Normalized Linear Complexity of Multi-sequences

Abstract

Asymptotic behavior of the normalized linear complexity $\frac{L_{\b{s}}(n)}{n}$ of a multi-sequence s̱ is studied in terms of its multidimensional continued fraction expansion, where $L_{\b{s}}(n)$ 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 $\lim \sup _{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$ together with a lower bound, and a formula for $\lim \inf_{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$ together with an upper bound are given. A necessary and sufficient condition for the existence of $\lim_{n\rightarrow\infty}\frac{L_{\b{s}}(n)}{n}$ is also given.