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

摘要

Some results on the k-normal elements and k-normal polynomials over finite fields are given in the recent literature. In this paper, we show that a transformation x -> (x(p) - x(p)(-1) + 1)/(-x(p)(-1) + 1) can be used to produce an infinite sequence of irreducible polynomials over a finite field F-q of characteristic p. By iteration of this transformation, we construct the k-normal polynomials of degree np(u) in F-q[x] starting from a suitable initial k-normal polynomial of degree n. We also construct an infinite sequence of k-normal polynomials using a certain quadratic transformation over F-2(s).