Generalized wordlength patterns and strength

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Xu and Wu (2001) defined the generalized wordlength pattern (A₁,..., Aₖ) 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 A₁ =... = Aₜ = 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.