Constructions Of (?) Designs Based On Singular Linear Spaces And Singular Symplectic Spaces

As a speical class of finite incidence incidence structures,(?)designs not only share most properies of partial geometries but also have close connection with strongly regular graphs and partilly balanced designs.In addition,antiflags of (?)designs can be used to constructure directed strongly regular graphs.Thus,constructions of (?)designs play a significant role in studying partial geometries,directed strong regular graphs and partilly balanced designs.Moreover,geometry of classical groups over finite fields are widely used to construct block designs because of its good combination structures.In this paper,four class of (?)designs are constructed by using subspaces of singular linear spaces and subspaces of singular symplectic spaces in sequence.Furthermore,some directed strongly regular graphs are given by using those (?)designs.1.Three classes of 11/2-designs are constructed by using subspaces of type(1,0),(n+ s,s)and(2,0),(n + l-1,l-1)in singular linear spaces.Then these (?)designs yield four infinite families of directed strongly regular graphs.All parameters of (?)designs and directed strongly regular graphs are computed.Moreover,some concrete examples of (?)designs and directed strongly regular graphs are showed.2.One class of (?)designs is constructed by using subspaces of type(m1,0,m1-1),(v+l,0,l)in singular symplectic spaces,then these (?)designs yield two infinite families of directed strongly regular graphs.All parameters of (?)designs and directed strongly regular graphs are computed.Moreover,some concrete examples of (?)designs and directed strongly regular graphs are showed.