Symplectic Groups, Symmetric Designs, and Line Ovals*

Let I7 = Sp(2m, 2) and I’ = X7, where Z is the translation group of the affine space AG(2m, 2). 17 acts 2-transitively on the cosets of each orthogonal subgroup G@(2m, 2), E = *l, and r has a second class of subgroups isomorphic to 17 ([lo, pp. 236, 2401, [6], and [14]). By considering a certain symmetric design P(2m) having r as its full automorphism group, we will prove these results. The symmetric designs 9+(2m) will be studied and characterized in terms of an interesting property concerning the symmetric difference of distinct blocks. Other properties and characterizations of these desings will also be given, and an application made to rank 3 linear groups. There are several ways to construct Yf(2m). One way is in terms of difference sets [3, p. 1081; from this point of view, Ye(2m) has the unusual property of arising from difference sets in m + 1 nonisomorphic abelian groups. Using a description in terms of the incidence matrix, Block [l] observed that the automorphism group of F(2m) is 2-transitive. The present work was motivated by Block’s result. In the course of studying the full automorphism group Yf(2m), a description was found in terms of GOs(2m, 2) and Sp(2m, 2) (see Section 4). More recently, the designs P(2m) were discovered in terms of the latter description by A. Rudvalis (unpublished) and Cameron and Seidel [2]. Yet another description of the designs P(2m) arises from using suitable dual ovals in desarguesian and Liineburg-Tits planes. The construction in terms of desarguesian planes was obtained by the author [3, p. 951 at the same time that the incidence matrix and symplectic group descriptions were first considered; however, it was not originally known that the descriptions produced the same designs. The construction from the Liineburg-Tits planes requires the use of the designs to prove a new property of the planes. We note