Pairs of $r$-primitive and $k$-normal elements in finite fields

Abstract

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