Optimality Of Symmetric L₂- Discrepancy

As a robust space filling design, uniform design has been widely used in sci-entific experiments, its theory is more systematic and thorough. To quantify the uniformity of design, there are various discrepancies defined, which measure the uniformity of a set of points on the unite hypercube. These discrepancies play a vital role in both the uniform design theory and in quasi-Monte Carlo methods. Each discrepancy has its own advantages and weakness. In this paper, we sum-marize and analyze the weakness and advantages of commonly used discrepancies, such as the L2-star discrepancy, the center L2-discrepancy(CD) and the wrap-around L2-discrepancy(WD). And then we studied the optimality of symmetric L2-discrepancy (SD). Find that as a measurement of the uniformity of the experi-mental design, SD has its own characteristics and advantages in terms of the intuitive view and the cover frequency. Other properties of SD are also studied, for example, The various forms of expression of SD; The relationships with other design criteria, we get the approximate equivalence between SD, orthogonal B-criteria and GMA criteria; Finally, a new proof of the lower bound are given based on the coincidence number forms of expression, and on this basis, we extended it to the 2,3 levels U-type design’s lower bound. This provides the theoretical support for searching and constructing the uniform designs under SD.