Existence of primitive normal pairs over finite fields with prescribed subtrace

Abstract

Given positive integers $q,n,m$ and $a\in\mathbb{F}_{q}$, where $q$ is an odd prime power and $n\geq 5$, we investigate the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{STr}_{q^n/q}(\epsilon)=a$, where $f(x)=\frac{f_{1}(x)}{f_{2}(x)}\in\mathbb{F}_{q^n}(x)$ is a rational function together with deg$(f_{1})+$deg$(f_{2})=m$ and $\mathrm{STr}_{q^n/q}(\epsilon) = \sum_{0\leq i<j\leq n-1}^{}\epsilon^{q^i+q^j}$. Finally, we conclude that for $m=2$, $n\geq 6$ and $q=7^k$; $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except at most $11$ choices.