Existence of primitive $2$-normal elements in finite fields (2007.11169v2)

Abstract: An element $\alpha \in \mathbb{F}{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}={\alpha, \alpha^q, \alpha^{q^2}, \cdots, \alpha^{q^{n-1}}}$ forms a basis of $\mathbb{F}{q^n}$ as a vector space over $\mathbb{F}q$. It is well known that $\alpha \in \mathbb{F}{q^n}$ is normal over $\mathbb{F}q$ if and only if $g{\alpha}(x)=\alpha x^{n-1}+\alpha^q x^{n-2}+ \cdots + \alpha^{q^{n-2}}x+\alpha^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $\alpha \in \mathbb{F}{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g{\alpha}[x]$ and $x^n-1$ in $\mathbb{F}{q^n}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}{q^n}$ over $\mathbb{F}q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}{q^n}$.