Subquadrangle m‐regular systems on generalized quadrangles

We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set ? of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of ?. If m is one half of the total number of subquadrangles on a point, we call ? a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface ??(3, q²), q odd, and two infinite families of symplectic subquadrangle hemisystems of ??₃(q²), q even. Some sporadic examples of symplectic subquadrangle regular systems of ??(3, q²) are also presented. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:28‐41, 2010