While the minimum aberration criterion is popular for selecting good designs with qualitative factors under an ANOVA model, the minimum $\beta$-aberration criterion is more suitable for selecting designs with quantitative factors under a polynomial model. In this paper, we propose the concept of wordlength enumerator to unify these two criteria. The wordlength enumerator is defined as an average similarity of contrasts among all possible pairs of runs. The wordlength enumerator is easy and fast to compute, and can be used to compare and rank designs efficiently. Based on the wordlength enumerator, we develop simple and fast methods for calculating both the generalized wordlength pattern and the $\beta$-wordlength pattern. We further obtain a lower bound of the wordlength enumerator for three-level designs and characterize the combinatorial structure of designs achieving the lower bound. Finally, we propose two methods for constructing supersaturated designs that have both generalized minimum aberration and minimum $\beta$-aberration.
Yu Tang. Hongquan Xu. "Wordlength enumerator for fractional factorial designs." Ann. Statist. 49 (1) 255 - 271, February 2021. https://doi.org/10.1214/20-AOS1955