# DESIGNS IN QUADRICS

#### Abstract

If (OMEGA) is the set of zeros in AG(2m, 2) of a nondegenerate quadratic form, then there is a group G of affine transformations isomorphic to the symplectic group Sp(2m, 2) which acts 2-transitively on (OMEGA). More generally, if m = r (.) s, then G has subgroups isomorphic to Sp(2r, 2('s)) and (GAMMA)Sp(2r, 2('s)) which act on (OMEGA) with rank at most 1 + 2('s-1). These subgroups induce association schemes on the points of (OMEGA) and provide means to construct families of balanced incomplete block designs and strongly regular graphs. One such family of designs takes as blocks the d-dimensional totally singular affine subspaces of (OMEGA). One particular (v, k, (lamda), r, b) design of this type (abbreviated to (v, k, (lamda))) has parameters (36, 8, 6) providing the smallest value of (lamda) known for any design with these values of v and k. The ovals of this design include an orbit under P(GAMMA)L(2, 8) which forms a (36, 6, 2) design, the first such design ever constructed without repeated blocks. Other families of designs consist of orbits of various arcs and ovals of AG(2, 2('m)) which lie inside an elliptic quadric. These designs have v = 1/2q(q - 1) points with q any power of 2. They have parameters (v, q - 1, q), (v, q + 1, q), and (v, q + 2, q + 2). In the first case the arcs partition the quadric implying that the design is resolvable. When q = 8 this determines a resolvable (28, 7, 8) design, the only design with these parameters known to be resolvable. The intersection properties between the lines of AG(2, 2('m)) and (OMEGA) are exploited to construct several families of designs. Examining the block orbit structure of these designs under Sp(2, 2('m)) enables us in a number of cases to find minimal subdesigns. One such family has parameters (v, q, 2q - 2), v as above. Families of designs are also constructed based on the points of a hyperbolic quadric in AG(2m, 2). Here v = 1/2q(q + 1) with q any power of two. These designs have parameters (v, q, q - 2) and (v, q, 2q - 6), the former providing, when q = 8, a second example of a (36, 8, 6) design. Additional designs are constructed, all based on the points of quadrics. In a number of cases it is possible to find subdesigns by examining the block orbit structure under a suitable subgroup of G. Included are appendices which list some of these constructions.

#### Subject Area

Mathematics

#### Recommended Citation

WERTHEIMER, MICHAEL AARON, "DESIGNS IN QUADRICS" (1986). *Dissertations available from ProQuest*. AAI8624036.

https://repository.upenn.edu/dissertations/AAI8624036