On the k-error linear complexity over $\mathbb{F}_p$ of Legendre and Sidelnikov sequences

Abstract

We determine exact values for the k-error linear complexity L _k over the finite field $\mathbb{F}_{p}$ of the Legendre sequence $\mathcal{L}$ of period p and the Sidelnikov sequence $\mathcal{T}$ of period p ^m − 1. The results are

$$ L_k(\mathcal{L}) =\left\{\begin{array}{ll} (p+1)/2, \quad 1 \le k \le (p-3)/2,\\ 0, \quad k\ge (p-1)/2, \end{array}\right.$$

$$L_k(\mathcal{T})\ge \min \left( \left( \frac{p+1}{2} \right)^{m}-1, \left \lceil \frac{p^m-1}{k+1} \right \rceil - \left(\frac{p+1}{2} \right)^{m} + 1 \right)$$

for 1 ≤ k ≤ (p ^m − 3)/2 and $L_k(\mathcal{T}) = 0$ for k≥ (p ^m − 1)/2. In particular, we prove

$$L_k(\mathcal{T}) = \left(\frac{p+1}{2} \right)^{m}-1,\quad 1\le k\le \frac{1}{2}\left(\frac{3}{2}\right)^{m}-1.$$