About $r$- primitive and $k$-normal elements in finite fields

Abstract

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $\alpha \in {\mathbb{F}}_{q^n}$ is $k$-normal over ${\mathbb{F}}_q$ if the greatest common divisor of the polynomials $g_{\alpha }(x)=\alpha x^{n-1}+{\alpha }^q{x}^{n-2}+\dots +{\alpha }^{q^{n-2}}x+{\alpha }^{q^{n-1}}$ and $x^n-1$ in ${\mathbb{F}}_{q^n}[x]$ has degree $k$, generalizing the concept of normal elements (normal in the usual sense is $0$-normal). In this paper we discuss the existence of $r$-primitive, $k$-normal elements in ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$, where an element $\alpha \in {\mathbb{F}}_{q^n}^{\ast }$ is $r$-primitive if its multiplicative order is $\frac{q^n-1}{r}$. We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic $11$.