Lower bounds for centered and wrap-around L2-discrepancies and construction of uniform designs by threshold accepting

We study the uniformity of two- and three-level U-type designs based on the centered and wrap-around L₂-discrepancies. By analyzing the known formulae, we find it possible to reexpress them as functions of column balance, and also as functions of Hamming distances of the rows. These new representations allow to obtain two kinds of lower bounds, which can be used as bench marks in searching uniform U-type designs. An efficient updating procedure for the local search heuristic threshold accepting is developed based on these novel formulations of the centered and wrap-around L₂-discrepancies. Our implementation of this heuristic for the two- and three-level case efficiently generates low discrepancy U-type designs. Their quality is assessed using the available lower bounds.