On the existence of primitive normal elements of rational form over finite fields of even characteristic

Let q be an even prime power and m >= 2 an integer. By F-q, we denote the finite field of order q and by F-qm its extension of degree m. In this paper, we investigate the existence of a primitive normal pair (alpha, f(alpha)), with f(x) = ax(2) + bx+c/dx+e is an element of F-qm(x) where the rank of the matrix F = (a 0 b d c e) is an element of M-2x3(F-qm) is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic" possess such elements, except for (1 0 1 1 0 0) if q = 2 and m is odd, and then we provide an explicit small list of possible and genuine exceptional pairs (q, m).