t-Designs in Projective Spaces

Delsarte, Goethals and Seidel originated the idea of t-designs of points on a sphere in R" by analogy with combinatorial r-designs [18]. These spherical t-designs have interesting combinatorial properties and relate to several mathematical disciplines [20]. The idea was extended by Neumaier [31] to his concept of r-designs in Delsarte spaces, the latter being so named by an analogy with the Q-polynomial association schemes of Delsarte (see [16, 30, 31]). In Section 2 we derive in a unified way the related special functions for Delsarte spaces IFP"-l (Section 1), which are projective spaces over IF =R, C, the quaternions IHI, or the Cayley numbers (or octonions) O. In terms of the dimension of IF over R, and for "distance sets" of arbitrary cardinality, we determine absolute bounds (Section 3) then special bounds and indicator coefficients [31] for r-designs (Sections 4 and 5). We give examples, some new some old, mainly for IF = C, IHI, 0 (the case IF = R is thoroughly studied in [17, 18, 20]). We include the generalized hexagon [7] on 819 points, with parameters (2, 8), as a 5-design in the Cayley Plane OP (Example 10 and 5.3,5.4) meeting the absolute bound. Several examples related to Leech's Lattice appear (Examples 11,27,28, Remark 5.4).