Connection between uniformity and orthogonality for symmetrical factorial designs

In this paper, we study the issue of uniformity in symmetrical fractional factorial designs. The discrete discrepancy (Biometrika 89 (2002) 893; Metrika 58 (2003) 279; Metrika 60 (2004) 59) is employed as a measure of uniformity. Although there are some emerging literature for connecting uniformity with orthogonality, less attention has been given to this issue for more than three-level fractional factorials and asymmetric fractional factorials. This paper discusses this issue for general symmetric fractional factorials. We derive results connecting uniformity and orthogonality and show that these criteria agree quite well, which provide further justifiable interpretation for some criteria of orthogonality by the consideration of uniformity. In addition, we also point that two measures of orthogonality in the literature (Fang, Hickernell, Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Springer, Berlin, 2002; J. Complexity 19 (2003) 692) are equivalent and derive now a lower bound of the discrete discrepancy.