Pairs of r-Primitive and k-Normal Elements in Finite Fields

Let $${\mathbb {F}}_{q^n}$$ be a finite field with $$q^n$$ elements and r positive divisor of $$q^n-1$$ . An element $$\alpha \in {\mathbb {F}}_{q^n}^*$$ is called r-primitive if its multiplicative order $$(q^n-1)/r$$ Also, k-normal over {F}}_q$$ the greatest common polynomials $$g_{\alpha }(x) = \alpha x^{n-1}+ ^q x^{n-2} + \ldots ^{q^{n-2}}x ^{q^{n-1}}$$ $$x^n-1$$ in {F}}_{q^n}[x]$$ has degree k. These concepts generalize ideas primitive normal elements, respectively. In this paper, we consider non-negative integers $$m_1,m_2,k_1,k_2$$ , $$r_1,r_2$$ rational functions $$F(x)=F_1(x)/F_2(x) {F}}_{q^n}(x)$$ $$\deg (F_i) \le m_i$$ for $$i\in \{ 1,2\}$$ satisfying certain conditions present sufficient existence $$r_1$$ -primitive $$k_1$$ -normal such that $$F(\alpha )$$ an $$r_2$$ $$k_2$$ Finally as example study case where $$r_1=2$$ $$r_2=3$$ $$k_1=2$$ $$k_2=1$$ $$m_1=2$$ $$m_2=1$$ $$n \ge 7$$