Number of $k$-normal elements over a finite field

TL;DR

This paper provides an explicit combinatorial formula for counting $k$-normal elements over finite fields, generalizing the concept of normal elements and solving an open problem from 2013.

Contribution

It introduces a combinatorial formula for the number of $k$-normal elements in finite fields, extending previous results and addressing an open problem.

Findings

Derived an explicit formula for $k$-normal elements count

Extended the concept of normal elements to $k$-normal elements

Solved an open problem from 2013 regarding $k$-normal elements

Abstract

An element $\alpha \in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $\alpha^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ is of the form $\{\alpha,\alpha^q, \ldots, \alpha^{q^{n-1}}\}$, where $\alpha \in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$. In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of k-normal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of $k$-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).