On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields

Abstract

Given ${\mathbb{F}}_{q^n}$, a field with $q^n$ elements, where $q$ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2\in \mathbb{N}$, $k_1,k_2\in \mathbb{N}\cup \{0\}$, a rational function $F=\frac{F_1}{F_2}$ in ${\mathbb{F}}_q[x]$ with deg($F_i$) $\le m_i$; $i=1,2,$ satisfying some conditions, and $a,b\in {\mathbb{F}}_q$, we construct a sufficient condition on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal element $\epsilon \in {\mathbb{F}}_{q^n}$ such that $F(\epsilon )$ is $r_2$-primitive, $k_2$-normal with ${\mathrm{Tr}}_{{\mathbb{F}}_{q^n}/{\mathbb{F}}_q}(\epsilon )=a$ and ${\mathrm{Tr}}_{{\mathbb{F}}_{q^n}/{\mathbb{F}}_q}({\epsilon }^{-1})=b$. For $m_1=10,m_2=11,r_1=3,r_2=2,k_1=2,k_2=1$, we establish bounds on $q$, for various $n$, to determine the existence of such elements in ${\mathbb{F}}_{q^n}$. Furthermore, we identify all such pairs $(q,n)$ excluding 10 possible values of $(q,n)$, in fields of characteristics 13.