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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 $mathbb{F}_q$: $$ sigma_n=left{ begin{array}{ll} 0,u0026 mathrm{if}quad n= 0, (1-chi(xi_n))/2,u0026mathrm{if}quad 1leq nu003c q, end{array} right. $$ for the ordered elements $xi_0,xi_1,ldots,xi_{q-1}in mathbb{F}_q$.  $sigma$ is Legendre sequence if $r=1$.  first contribution is to prove a lower bound on the linear complexity of $sigma$ for $rgeq 2$. 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 $ru003e2$ and leave it open.

complexity

On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-Error Linear Complexity of Binary Sequences Derived from the Discrete Logarithm in Finite Fields

On the k-Error Linear Complexity of Binary Sequences Derived from the Discrete Logarithm in Finite Fields