A primitive normal pair with prescribed prenorm

Abstract

For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{PN}_{q^n/q}(\epsilon)=a$, where $a\in\mathbb{F}_{q}$ is prescribed. Here $f={f_{1}}/{f_{2}}\in\mathbb{F}_{q^n}(x)$ is a rational function subject to some minor restrictions such that deg($f_{1}$)+deg($f_{2}$)$=m$ and $\mathrm{PN}_{q^n/q}(\epsilon) =\sum_{i=0}^{n-1}\Bigg(\underset{j\neq i}{\underset{0\leq j\leq n-1}{\prod_{}^{}}}\epsilon^{q^j}\Bigg)$. Finally, we conclude that for $m=3$, $n\geq 6$, and $q=7^k$ where $k\in\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except possibly $10$ choices at most.