On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields
Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in $\mathbb{F}_{q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
18.04.2023
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Subjects | |
Online Access | Get full text |
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Summary: | Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a
prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$,
$k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$
in $\mathbb{F}_{q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying some
conditions, and $a,b \in \mathbb{F}_{q}$, we construct a sufficient condition
on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal
element $\epsilon \in \mathbb{F}_{q^n}$ such that $F(\epsilon)$ is
$r_2$-primitive, $k_2$-normal with
$\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(\epsilon) = a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(\epsilon^{-1}) = b$. For
$m_1=10, \; m_2=11,\; r_1 = 3, \; r_2 = 2, \; k_1=2,\;k_2 = 1$, we establish
bounds on $q$, for various $n$, to determine the existence of such elements in
$\mathbb{F}_{q^{n}}$. Furthermore, we identify all such pairs $(q,n)$ excluding
10 possible values of $(q,n)$, in fields of characteristics 13. |
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DOI: | 10.48550/arxiv.2304.08749 |