On a class of 1-designs (Q1209289)

\textit{E. F. Assmus} and the author [ibid. 10, No. 4, 297-308 (1989; Zbl 0713.05019)] constructed a class of \((q^ 3+1,q+1,q+2)\)-designs by adding suitable blocks to the Ree unitals. This is analogous to a construction of \textit{G. Hölz} [Arch. Math. 37, 179-183 (1981; Zbl 0451.05015)] who started with the hermitian unitals. The Hölz designs can also be described as a one-point extension of the generalized quadrangles of Ahrens and Szekeres. Answering a question put forward by V. D. Tonchev and the reviewer, the author shows that the Assmus-Key designs cannot be obtained as extensions of generalized quadrangles (except for the case \(q=3\), where the two constructions coincide). In particular, this gives an elementary proof for the fact that these two families of designs are non-isomorphic. The proof rests on showing that the polynomial function \(x(x^{p^{n+1}}-x-1)\) cannot be a permutation polynomial of \(GF(p^{2n+1})\) for any prime \(p\).