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Traditional global stability measure for sequences is hard to determinebecause of large search space. We propose the $k$-error linear complexity witha zone restriction for measuring the local stability of sequences. Accordingly,we can efficiently determine the global stability by studying a local stabilityfor these sequences. For several classes of sequences, we demonstrate that the$k$-error linear complexity is identical to the $k$-error linear complexitywithin a zone, while the length of a zone is much smaller than the whole periodwhen the $k$-error linear complexity is large. These sequences have periods$2^n$, or $2^v r$ ($r$ odd prime and $2$ is primitive modulo $r$), or $2^vp_1^{s_1} \cdots p_n^{s_n}$ ($p_i$ is an odd prime and $2$ is primitive modulo$p_i$ and $p_i^2$, where $1\leq i \leq n$) respectively. In particular, wecompletely determine the spectrum of $1$-error linear complexity with any zonelength for an arbitrary $2^n$-periodic binary sequence.

On the stability of periodic binary sequences with zone restriction