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.

中文翻译：

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分数阶乘设计的字长枚举器

尽管最小像差准则在ANOVA模型下选择具有定性因子的良好设计时很受欢迎，但最小$ \ beta $-像差准则更适合在多项式模型下选择具有定量因子的设计。在本文中，我们提出了字长枚举器的概念以统一这两个标准。字长枚举器定义为所有可能的运行对之间对比度的平均相似度。字长枚举器易于计算，可以快速地对设计进行比较和排名。基于wordlength枚举器，我们开发了简单而快速的方法来计算广义wordlength模式和$ \ beta $ -wordlength模式。我们进一步为三级设计获得了字长枚举器的下界，并描述了实现下界的设计的组合结构。最后，我们提出了两种构造过饱和设计的方法，它们既具有广义的最小像差又具有最小的$ \ beta $-像差。