Existence of primitive normal pairs over finite fields with prescribed subtrace
K. Chatterjee, G. Kapetanakis, H. Sharma, S. K. Tiwari
TL;DR
This work studies the existence of primitive normal pairs $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ with a prescribed subtrace $\mathrm{STr}_{q^n/q}(\epsilon)$, where $f(x)$ is a rational function of total degree $m$. The authors formulate a counting framework using additive and multiplicative characters, freeness conditions, and trace constraints, and derive a main sufficient condition: if $q^{\frac{n}{2}-2} > 2m\,W(l)^2\,W(L)\,W(x^n-1)$, then a suitable primitive normal pair exists (membership in $\mathcal{D}_m$). They extend the approach with sieve methods (standard, prime, and modified prime sieves) to relax the bound and broaden applicability, introducing $\mathcal{S}$ and $\mathcal{M}$ parameters to optimize the sieve. A detailed numerical case study for $q=7^k$ and $m=2$ shows that $\mathcal{D}_2$ holds for all $q=7^k$, $n\ge 6$ except a small finite list of exceptions, with explicit parameter choices provided. The results advance understanding of primitive normal pairs with prescribed subtrace and demonstrate practical feasibility via computational sieving.
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 $(ε,f(ε))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such that $\mathrm{STr}_{q^n/q}(ε)=a$, 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 $\mathrm{STr}_{q^n/q}(ε) = \sum_{0\leq i<j\leq n-1}^{}ε^{q^i+q^j}$. 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.