On the stability of periodic binary sequences with zone restriction

Traditional global stability measure for sequences is hard to determine because of large search space. We propose the k-error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the k-error linear complexity is identical to the k-error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the k-error linear complexity is large. These sequences have periods $$2^n$$ 2 n , or $$2^v r$$ 2 v r (r odd prime and 2 is primitive modulo r), or $$2^v p_1^{s_1} \cdots p_n^{s_n}$$ 2 v p 1 s 1 ⋯ p n s n ($$p_i$$ p i is an odd prime and 2 is primitive modulo $$p_i^2$$ p i 2 , where $$1\le i \le n$$ 1 ≤ i ≤ n ) respectively. In particular, we completely determine the spectrum of 1-error linear complexity with any zone length for an arbitrary $$2^n$$ 2 n -periodic binary sequence.