Some identities on q n-m designs with application to minimum aberration designs

Chen and Hedayat and Tang and Wu studied and characterized minimum aberration 2ⁿ⁻ᵐ designs in terms of their complemetary designs. Based on a new and more powerful approach, we extend the study to identify minimum aberration qⁿ⁻ᵐ designs through their complementary designs. By using MacWilliams identities and Krawtchouk polynomials in coding theory, we obtain some general and explicit relationships between the wordlength pattern of a qⁿ⁻ᵐ design and that of its complementary design. These identities provide a powerful tool for characterizing minimum aberration qⁿ⁻ᵐ designs. The case of $q = 3$ is studied in more details.