Generalized resolution for orthogonal arrays

The generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion (Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of $R^2$ values (based on orthogonal coding) or sums of squared canonical correlations (based on arbitrary coding). Directly related to these results, we derive two versions of generalized resolution for qualitative factors, both of which are generalizations of the generalized resolution by Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann. Statist. 27 (1999) 1914-1926]. We provide a sufficient condition for one of these to attain its upper bound, and we provide explicit upper bounds for two classes of symmetric designs. Factor-wise generalized resolution values provide useful additional detail.