Existence of primitive normal pairs over finite fields with prescribed subtrace

TL;DR

This paper proves the existence of primitive normal pairs with a prescribed subtrace in finite fields, extending the understanding of such pairs for specific parameters and showing they exist in almost all cases.

Contribution

It establishes the existence of primitive normal pairs with a given subtrace over finite fields for certain parameters, including explicit exceptions, advancing finite field theory.

Findings

Existence of primitive normal pairs with prescribed subtrace for n ≥ 6 and q=7^k.

Almost all pairs (q,n) satisfy the existence condition, except at most 11 cases.

Results apply to rational functions with degree sum m=2.

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.