Generalized wordlength enumerator for asymmetrical designs

In this paper, we extend the study of wordlength enumerator to asymmetrical designs, and unify the generalized $\alpha$- and $\beta$-wordlength pattern with generalized wordlength enumerator. The generalized wordlength enumerator is easy and fast to compute, and can effectively compare and rank symmetrical and asymmetrical designs. A lower bound of generalized wordlength enumerator is provided, and can be used as a benchmark for searching and constructing optimal designs. Numerical results are also provided, which lend further support to our theoretical findings.