\(s_1^{n_1}\)-\(s_2^{n_2}\) space-filling designs and projections based on level permutations

Level permutations of factors can alter geometrical structures and statistical properties of designs. In this paper, we study the space-filling properties of asymmetric designs and their projections under two commonly used space-filling measures, maximin distance and maximum projection. When all possible level permutations of factors are considered, the analytic connections between the measure defined by maximin distance or the measure defined by maximum projection and generalized wordlength pattern are built for asymmetric designs, respectively. The generalized minimum aberration designs tend to have good space-filling properties in terms of distance and projection in general. Furthermore, several novel relationships between distance distribution and generalized wordlength pattern are derived. Finally, some asymmetric designs with good space-filling properties and less aberration are obtained by level permutations of factors for practical use.