On a Class of 1-Designs

It is shown that the derived 1-(q3, q, q, + 2) designs obtained from the 2-(q3 + 1, q + 1, q + 2) designs of Assmus and Key [2], where q = 32n + 1, are not generalized quadrangles unless n = 0. The proof involves showing that, for q = p2n + 1, the polynomial function x (xs - x - 1), where s = pn + 1 , is not a permutation polynomial of the field Fq for any prime p and n ⩾ 1.