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

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 $\alpha \in {\mathbb{F}}_{q^n}^{\ast }$ 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 $\gcd (\alpha x^{n-1}+{\alpha }^q{x}^{n-2}+\dots +{\alpha }^{q^{n-2}}x+{\alpha }^{q^{n-1}},x^n-1)$ in ${\mathbb{F}}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of elements in arithmetic progressions $\{\alpha ,\alpha +\beta ,\alpha +2\beta ,\dots \alpha +(m-1)\beta \}\subset {\mathbb{F}}_{q^n}$ 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 $i\in \{1,\dots ,m\}$.

To appear in Communications in Algebra. arXiv admin note: substantial text overlap with arXiv:2210.11504