On r-primitive k-normal elements over finite fields

Let r, n be positive integers, k be a non-negative integer and q be any prime power such that r|qn−1. An element α of the finite field Fqn is called an r-primitive element, if its multiplicative order is (qn−1)/r and it is called a k-normal element over Fq, if the degree of the greatest common divisor of the polynomials mα(x)=∑i=1nαqi−1xn−i and xn−1 is k. In this article, we discuss the existence of an element α∈Fqn which is both r-primitive and k-normal over Fq. In particular, we show that there exists an element α∈Fqn, which is both 2-primitive and 2-normal over Fq if and only if q is an odd prime power and either n≥5 and gcd(q3−q,n)≠1 or n=4 and q≡1(mod4).