About r-primitive and k-normal elements in finite fields

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements: an element alpha is an element of F-qn is k-normal over F-q if the greatest common divisor of the polynomials g(alpha)(x) = alpha x(n-1)+alpha(q) x(n-2)+ ... +alpha(qn-2) x+alpha(qn-1) and x(n)-1 in F-qn[x] has degree k, generalizing the concept of normal elements (normal in the usual sense is 0-normal). In this paperwe discuss the existence of r-primitive k-normal elements in F-qn over F-q, where an element alpha is an element of F-qn* is r-primitive if its multiplicative order is q(n)-1/r. We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic 11.