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arXiv · 2211.02114

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

Josimar J. R. Aguirre·Abílio Lemos·Victor G. L. Neumann·Sávio Ribas
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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 $i \in \{1, \dots, m\}$.

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Josimar J. R. Aguirre, Abílio Lemos, Victor G. L. Neumann, Sávio Ribas. 2022-11-03. $r$-primitive $k$-normal elements in arithmetic progressions over finite fields. https://arxiv.org/abs/2211.02114

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