Generalized wordlength patterns and strength

TL;DR

This paper provides a character-theoretic proof that the strength of a fractional factorial design can be characterized by the generalized wordlength pattern, extending the result beyond cyclic groups to nonabelian groups under certain conditions.

Contribution

It offers a direct character-theoretic proof of the relationship between wordlength patterns and design strength, generalizing previous results to nonabelian groups.

Findings

Proof applies to non-cyclic groups with specific conditions

Strength is characterized by vanishing initial wordlength pattern components

Extends the theoretical framework for analyzing factorial designs

Abstract

Xu and Wu (2001) defined the \emph{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 $A_i$ 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.