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

Abstract

Recently, the [Formula: see text]-normal elements over finite fields are defined and characterized by Huczynska et al. In this paper, we give a new characterization of [Formula: see text]-normal elements and define [Formula: see text]-normal polynomials over finite fields. In what follows, we show that the problem of existence of a primitive 1-normal element in [Formula: see text] over [Formula: see text], for all [Formula: see text] and [Formula: see text], which has been stated by Huczynska et al., is not satisfied. Furthermore, we extend a recursive method given by Kyuregyan for constructing an infinite family of [Formula: see text]-polynomials, to constructing an infinite sequence of [Formula: see text]-normal polynomials over [Formula: see text].