A primitive normal pair with prescribed prenorm

TL;DR

This paper establishes conditions for the existence of primitive normal pairs with prescribed prenorm in finite fields, extending previous results and providing nearly complete existence results for specific parameters.

Contribution

It introduces new sufficient conditions for primitive normal pairs with prescribed prenorm in finite fields, including nearly complete existence results for certain parameters.

Findings

Existence of primitive normal pairs under new conditions

Almost all pairs exist for specified parameters with few exceptions

Provides a framework for constructing such pairs 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.