On the $k$-error linear complexity of binary sequences derived from the discrete logarithm in finite fields

Let $q=p^r$ be a power of an odd prime $p$. We study binary sequences $\sigma=(\sigma_0,\sigma_1,\ldots)$ with entries in $\{0,1\}$ defined by using the quadratic character $\chi$ of the finite field $\mathbb{F}_q$: $$\sigma_n=\left\{ \begin{array}{ll} 0,& \mathrm{if}\quad n= 0,\\ (1-\chi(\xi_n))/2,&\mathrm{if}\quad 1\leq n< q, \end{array} \right.$$ for the ordered elements $\xi_0,\xi_1,\ldots,\xi_{q-1}\in \mathbb{F}_q$. The $\sigma$ is Legendre sequence if $r=1$. Our first contribution is to prove a lower bound on the linear complexity of $\sigma$ for $r\geq 2$. The bound improves some results of Meidl and Winterhof. Our second contribution is to study the $k$-error linear complexity of $\sigma$ for $r=2$. It seems that we cannot settle the case when $r>2$ and leave it open.