On the $k$-error linear complexity of binary sequences derived from polynomial quotients

TL;DR

This paper analyzes the $k$-error linear complexity of binary sequences derived from polynomial quotients, providing exact values under certain conditions and demonstrating their robustness in terms of error linear complexity.

Contribution

It determines the exact $k$-error linear complexity of these sequences over finite fields, including cases with primitive roots, and highlights their strong error linear complexity properties.

Findings

Exact $k$-error linear complexity values are obtained for sequences over $_2$ and $_p$.

Sequences exhibit good error linear complexity, indicating robustness.

Results depend on the primitive root assumption and parameters $w$ and $k$.

Abstract

We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0, $$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field $\F_2$ for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field $\F_p$ for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.