Lower bounds for wrap-around L₂-discrepancy and constructions of symmetrical uniform designs

The wrap-around L2-discrepancy has been used in quasi-Monte Carlo methods, especially in experimental designs. In this paper, explicit lower bounds of the wrap-around L2-discrepancy of U-type designs are obtained. Sufficient conditions for U-type designs to achieve their lower bounds are given. Taking advantage of these conditions, we consider the perfect resolvable balanced incomplete block designs, and use them to construct uniform designs under the wrap-around L2-discrepancy directly. We also propose an efficient balance-pursuit heuristic, by which we find many new uniform designs, especially with high levels. It is seen that the new algorithm is more powerful than existing threshold accepting ones in the literature.