Kummer configurations and designs embedded in planes (Q1823951)

Let PG(2,q) denote the finite projective plane over the field with q elements. In PG(2,5), \textit{E. F. Assmus} [Planes and biplanes revisited, J. Comb. Theory, Ser. A (to appear)] observed the existence of 16 points and 16 conics such that any two distinct conics meet in precisely two points, thereby constituting a (16,6,2)-design. In the present paper, the author shows that in PG(2,q), \(q>3\) and q odd, there is a set Z of 16 points and a set W of 16 conical sets so that Z and the subsets W form a (16,6,2)-design.