Generalized wordlength patterns and strength

Xu and Wu (2001) defined the generalized wordlength pattern ( A 1 , … , A k ) of an arbitrary fractional factorial design (or orthogonal array) on k factors. They gave a coding-theoretic proof of the property that the design has strength t if and only if A 1 = ⋯ = A t = 0 . The quantities Ai are defined in terms of characters of cyclic groups, and so one might seek a direct character-theoretic proof of this result. We give such a proof, in which the specific group structure (such as cyclicity) plays essentially no role. Nonabelian groups can be used if the counting function of the design satisfies one assumption, as illustrated by a couple of examples.