$r$-primitive $k$-normal elements in arithmetic progressions over finite fields

TL;DR

This paper investigates the existence of elements in finite field arithmetic progressions that are both $r$-primitive and $k$-normal, providing asymptotic and specific results for certain parameters.

Contribution

It introduces new results on the existence of $r$-primitive and $k$-normal elements in arithmetic progressions over finite fields, including asymptotic and specific cases.

Findings

Asymptotic existence results for general $k, r_i$

Concrete existence results for $k = r_i = 2$

Conditions under which such elements are guaranteed to exist

Abstract

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 $i \in \{1, \dots, m\}$.