Some notes on the $k$-normal elements and $k$-normal polynomials over finite fields

TL;DR

This paper explores the properties and characterization of k-normal elements and polynomials over finite fields, generalizes existing theorems for checking normality, and introduces a recursive method for constructing higher-degree 1-normal polynomials.

Contribution

It provides a new characterization of k-normal elements using a generalization of Schwartz's theorem and presents a recursive construction method for 1-normal polynomials.

Findings

Characterization of k-normal elements using a generalized Schwartz's theorem.

The problem of primitive 1-normal elements is generally not satisfied.

A recursive method for constructing higher-degree 1-normal polynomials over finite fields.

Abstract

Recently, the $k$-normal element over finite fields is defined and characterized by Huczynska et al.. In this paper, the characterization of $k$-normal elements, by using to give a generalization of Schwartz's theorem, which allows us to check if an element is a normal element, is obtained. In what follows, in respect of the problem of existence of a primitive 1-normal element in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, for all $q$ and $n$, had been stated by Huczynska et al., it is shown that, in general, this problem is not satisfied. Finally, a recursive method for constructing $1$-normal polynomials of higher degree from a given $1$-normal polynomial over $\mathbb{F}_{2^m}$ is given.