The prime power conjecture is the most prominent open problem in combinatorial design theory. Its solution would have far-reaching consequences in areas as diverse as coding theory, statistical experiment design and finite geometry. This project will develop the theoretical tools needed to attack the conjecture and other closely related problems. The patterns it uncovers will reveal new codes for communication and new, more efficient designs for statistical experiments. The project may also prove in some cases that there are no better codes or designs than what we currently use. In the process it will dramatically increase our knowledge base in combinatorics, which is a branch of mathematics of crucial importance in the computer age.