About r-primitive and k-normal elements in finite fields

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements: an element $$\alpha \in {\mathbb {F}}_{q^n}$$ is over $${\mathbb {F}}_q$$ if greatest common divisor polynomials $$g_{\alpha }(x)= \alpha x^{n-1}+\alpha ^qx^{n-2}+\cdots +\alpha ^{q^{n-2}}x+\alpha ^{q^{n-1}}$$ $$x^n-1$$ in {F}}_{q^n}[x]$$ has degree k, generalizing normal elements (normal usual sense 0-normal). this paper we discuss existence r-primitive {F}}_{q}$$ , where {F}}_{q^n}^*$$ its multiplicative order $$\frac{q^n-1}{r}$$ . We provide many general results about class work a numerical example finite fields characteristic 11.