Some results on two-level regular designs with general minimum lower-order confounding

Zhang et al. (2008) introduced an aliased effect-number pattern (AENP) for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. By using a finite projective geometric formulation, Zhang and Mukerjee (2009a) characterized GMC designs via complementary designs for general s-level case, and to find GMC designs, for some special cases they proved a result that a design T can have GMC only if T ¯ is contained in a specific flat. In this paper, we first generalize the result to general cases for s=2. Then, we prove that, for any given n and m, a GMC design minimizes A3, the first term of the wordlengh pattern of regular 2n−m designs. Furthermore, we find out the unique optimal confounding structure between main effects and two-factor interactions, and prove that minimizing A3 is a sufficient and necessary condition for a regular design to have the structure.