  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.
