Combinatorial Properties And Constructions Of Uniform Designs

The connection between the experimental designs and combinatorial designs has a long history. The present thesis aims to investigate corresponding combinatorial properties for uniform designs under some frequently used discrepancies. It also provides construction methods for uniform designs by using various known or new-established combinatorial configurations.According to three different types of discrepancies, i.e., the discrete discrepancy, the wrap-around L₂-discrepancy and the centered L₂-discrepancy, the main part of the thesis is divided into three chapters. Chapter 2 generalizes the known combinatorial properties required for uniform designs under the discrete discrepancy and uses various combinatorial configurations including resolvable partially pairwise balanced designs, resolvable packings and coverings and t-designs to obtain corresponding uniform designs. In Chapter 3 and Chapter 4, we deal with the combinatorial properties and also construction methods for uniform designs under the wrap-around L₂-discrepancy and the centered L₂-discrepancy, respectively. A series of lower bounds for the discrepancies and corresponding conditions for achieving these lower bounds are explicitly given. Some special combinatorial configurations including the perfect resolvable balanced incomplete block design are defined to construct uniform designs. In the last chapter, we also propose a balanced pursuit algorithm to search low-discrepancy design under any given discrepancy. This algorithm is combined with the combinatorial properpies discribed in previous chapters and seems to be efficient. As a result, in many cases we can find fairly satisfactory designs by implementing it.