Primitive Normal Polynomials Over Finite Fields

In this note we significantly extend the range of published tables of primitive normal polynomials over finite fields. For each with , we provide a primitive normal polynomial of degree n over . Moreover, each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over . The roots of such a polynomial generate a primitive normal basis of over , and so are of importance in many computational problems. We also raise several conjectures concerning the distribution of such primitive normal polynomials, including a refinement of the primitive normal basis theorem.