A primitive normal pair with prescribed prenorm

Abstract

For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n\ge 5$, we establish a condition sufficient to ensure 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 ${\mathrm{PN}}_{q^n/q}(\hat{I}\mu )=a$, 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 ${\mathrm{PN}}_{q^n/q}(\hat{I}\mu )={\sum }_{i=0}^{n-1}\left(\underset{j\ne i}{{\prod }_{}^{}_{0\le j\le n-1}}\hat{I}{\mu }^{q^j}\right)$. Finally, we conclude that for $m=3$, $n\ge 6$, 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.

arXiv admin note: substantial text overlap with arXiv:2405.11463