Some identities on $q\sp {n-m}$ designs with application to minimum aberration designs

Abstract

Chen and Hedayat and Tang and Wu studied and characterized minimum aberration $2^{n-m}$ designs in terms of their complemetary designs. Based on a new and more powerful approach, we extend the study to identify minimum aberration $q^{n-m}$ 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 $q^{n-m}$ design and that of its complementary design. These identities provide a powerful tool for characterizing minimum aberration $q^{n-m}$ designs. The case of $q = 3$ is studied in more details.