# On the stability of periodic binary sequences with zone restriction

## TL;DR

This paper introduces a zone-restricted $k$-error linear complexity measure to efficiently analyze the local and global stability of periodic binary sequences, especially for sequences with specific period structures.

## Contribution

It proposes a novel zone restriction approach for $k$-error linear complexity, enabling efficient stability analysis and complete spectrum determination for certain classes of sequences.

## Key findings

- $k$-error linear complexity equals zone-restricted complexity for specific sequence classes.
- Zone length can be much smaller than the period when the complexity is large.
- Complete spectrum of 1-error linear complexity is determined for $2^n$-periodic sequences.

## Abstract

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. Accordingly, we can efficiently determine the global stability by studying a local stability for these 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 $2^n$, or $2^v r$ ($r$ odd prime and $2$ is primitive modulo $r$), or $2^v p_1^{s_1} \cdots p_n^{s_n}$ ($p_i$ is an odd prime and $2$ is primitive modulo $p_i$ and $p_i^2$, where $1\leq i \leq n$) respectively. In particular, we completely determine the spectrum of $1$-error linear complexity with any zone length for an arbitrary $2^n$-periodic binary sequence.