Recursive constructions of k-normal polynomials using some rational transformations over finite fields

Abstract

Some results on the [Formula: see text]-normal elements and [Formula: see text]-normal polynomials over finite fields are given in the recent literature. In this paper, we show that a transformation [Formula: see text] can be used to produce an infinite sequence of irreducible polynomials over a finite field [Formula: see text] of characteristic [Formula: see text]. By iteration of this transformation, we construct the [Formula: see text]-normal polynomials of degree [Formula: see text] in [Formula: see text] starting from a suitable initial [Formula: see text]-normal polynomial of degree [Formula: see text]. We also construct an infinite sequence of [Formula: see text]-normal polynomials using a certain quadratic transformation over [Formula: see text].