Space-filling designs are routinely employed in computer experiments, and common criteria for selecting such designs are either distance- or discrepancy-based. Recently, Tian and Xu introduced a minimum aberration-type criterion known as the Space-Filling Pattern (SFP), which plays a crucial role in classifying and ranking strong orthogonal arrays, a well-established category of space-filling designs. Designs that perform well under the SFP tend to exhibit stratifications across a variety of grids. However, SFP does not distinguish between grids of different dimensions. To address this, we propose a refined version of the SFP, named the Stratification Pattern (SP). Using the chi-characteristics, we provide a justification for both the SFP and SP. Next, our focus shifts to the regular designs of a prime square level. We show that the SP-optimal designs can be found by counting the different types of words of the same lengths. This result allows for a complete search for the SP-optimal designs, especially for those of small sizes.  

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