Applications Of Geometry Of Classical Groups In Combinatorics And Designs

The theory of geometry of classical groups over finite fields has developed thoroughly. And there are lots of important applications of this theory:such as association schemes, lattices, graphs, designs, codes etc. In this thesis, we concern the association schemes, the lattices, the graphs and the pooling designs. The main results of this thesis are:1. As a generalization of the classical dual polar schemes, using the matrix method, we construct some families of new association schemes with the (m, k)-isotropic subspaces in singular classical spaces, compute the parameters of these schemes, determine the character tables of these schemes and study the graphs associated with these schemes.2. We characterize the elements in the lattices generated by two orbits of subspaces in finite singular symplectic and unitary spaces, and discuss their geometric structures.3. Using the subspaces of symplectic spaces over finite fields, we construct new Deza graphs, and determine the spectra, the chromatic numbers and the independence numbers of these graphs.4. Using the subspaces of classical spaces over finite fields, we construct some new pool-ing designs which have more useful applications in DNA library screening.