Prescribed automorphism groups and two problems in Galois geometries

We study two combinatorial design problems: finding small $t$-fold blocking sets in $PG(2,q)$ for $q$ prime, and finding large caps in $PG(n,q)$. These problems are combinatorially difficult, but their problem sizes can be reduced by prescribing a group of automorphisms in order to take advantage of the anticipated symmetry of the solution sets. Using this method, a new family of 2-fold blocking sets of size $3q+1$ was discovered. These new sets have the additional property that their complement is also a 2-fold blocking set. For caps, a backtracking algorithm with pruning was developed to solve the resulting 0-1 int eger linear programming problem, and was implemented using C. This algorithm takes advantage of the small size of the right-hand side of the constraints in the LP problem. Also presented is an introduction to finite projective geometries and group actions and the application of group actions to projective geometries.