Grid representations for 2IVm-(m-k) join designs containing maximum number of clear two-factor interactions

This paper shows that the join designs [Butler, N.A., 2005. Classification of efficient two-level fractional factorial design of resolution IV or more. J. Statist. Plann. Inference 131, 145–159] include all 2IVm-(m-k) designs with the maximum number of clear two-factor interactions (2FIs) for k=5,6,7k=5,6,7. The designs are shown to have concise grid representations which provide simple interpretations of the 2FIs in a 2IVm-(m-k) design. In addition, a better upper bound on the maximum number of clear 2FIs in a 2IVm-(m-k) design than the one obtained by Tang et al. [2002. Bounds of the maximum members of clear factor interactions for 2m-p2m-p designs of resolution III and IV. Canad. J. Statist. 30, 127–136] is given.