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 $n\ge 5$, we investigate the existence of a primitive normal pair $(\hat{I}\mu ,f(\hat{I}\mu ))$ in ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ such that ${\mathrm{STr}}_{q^n/q}(\hat{I}\mu )=a$, 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 ${\mathrm{STr}}_{q^n/q}(\hat{I}\mu )={\sum }_{0\le i<j\le n-1}^{}\hat{I}{\mu }^{q^i+q^j}$. Finally, we conclude that for $m=2$, $n\ge 6$ and $q=7^k$; $k\in \mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except at most $11$ choices.