  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.
