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 p" < 1050 with P < 97, we provide a primitive normal polynomial of degree n over Fp. Moreover, each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over Fp. The roots of such a polynomial generate a primitive normal basis of Fpn over Fp, 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.