Examples of combinatorial designs

In this thesis we investigate examples of combinatorial designs. We begin by considering a (16, 6, 2) configuration of points and planes in projective three space over fields of characteristic not 2. We define isomorphic configurations and discuss several subconfigurations arising from this (16, 6, 2) design. We also count the number of isomorphic designs with the same point set to this particular (16, 6, 2) one.; In Chapter 4, we consider designs and quadratic forms. In particular, we give an extension theorem relating tactical configurations coming from nondegenerate quadratic forms in an odd number of indeterminates over $F2$ to a class of symmetric (v, k, λ) configurations. In particular, one such extended design is isomorphic to the original (16, 6, 2) configuration.; Next, we give background information on orthogonal groups over finite fields of characteristic 2. For a certain symmetric (v, k, λ) configuration, we count the number of symmetric tactical subconfigurations of a given type. Then we construct symmetric tactical configurations coming from nondegenerate quadratic forms over fields of characteristic not 2.; Lastly, we look at certain designs coming from finite geometry. We define the Möbius 8₄ configuration and show it is in fact isomorphic to the 8₄ design constructed in Chapter 4. We then consider designs in both affine and projective geometry over finite fields. We end with an example of a design coming from ovals in PG(2, 4) and relate it to the original (16, 6, 2) configuration.