rr-Primitive kk-Normal elements in Arithmetic Progressions over Finite Fields

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $\alpha \in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a non-negative integer $k$, the element $\alpha \in \mathbb{F}_{q^n}$ is \textit{$k$-normal} over $\mathbb{F}_q$ if $\gcd(\alpha x^{n-1}+ \alpha^q x^{n-2} + \ldots + \alpha^{q^{n-2}}x + \alpha^{q^{n-1}} , x^n-1)$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of elements in arithmetic progressions $\{\alpha, \alpha+\beta, \alpha+2\beta, \ldots\alpha+(m-1)\beta\} \subset \mathbb{F}_{q^n}$ with $\alpha+(i-1)\beta$ being $r_i$-primitive and at least one of the elements in the arithmetic progression being $k$-normal over $\mathbb{F}_q$. We obtain asymptotic results for general $k, r_1, \dots, r_m$ and concrete results when $k = r_i = 2$ for $i \in \{1, \dots, m\}$.Comment: arXiv admin note: substantial text overlap with arXiv:2210.1150

On pairs of rr-primitive and kk-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, $n$ is positive integer. For $r \in \mathbb{N}$, $k \in \mathbb{N} \cup \{ 0 \}$, an element $\epsilon \in \mathbb{F}_{q^n}$ is said to be $r$-primitive if its multiplicative order is $\frac{q^n -1}{r}$ and it is referred to as $k$-normal if the greatest common divisor of the polynomial $\sum_{i=0}^{n-1} \epsilon^{q^i} x^{n-1-i}$ with $x^n -1$ has degree $k$ in $\mathbb{F}_{q^n}[x]$. In this article, 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 $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(\epsilon^{-1}) = b$. Further, for $m_1=10 , m_2=11$, we demonstrate an example showing the existence of 3-primitive, 2-normal element $\epsilon$ in $\mathbb{F}_{q^n}$ such that $F(\epsilon)$ is 2-primitive, 1-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 any prescribed $a,b \in \mathbb{F}_{q}$ except from possible 10 values of $(q,n)$ in field of characteristics 13