Grid representations for \(2^{m-(m-k)}_{\text{IV}}\) join designs containing maximum number of clear two-factor interactions (Q2475779)

This paper shows that the join designs [\textit{N. A. Butler}, Classification of efficient two-level fractional factorial design of resolution IV or more. ibid. 131, 145--159 (2005; Zbl 1067.62076)] include all \(2^{m-(m-k)}_{\text{IV}}\) designs with the maximum number of clear two-factor interactions (2FIs) for \(k=5,6,7.\) The designs are shown to have concise grid representations which provide simple interpretations of the 2FIs in a \(2^{m-(m-k)}_{\text{IV}}\) design. In addition, a better upper bound on the maximum number of clear 2FIs in a \(2^{m-(m-k)}_{\text{IV}}\) design than the one obtained by \textit{B. Tang} et al. [Bounds of the maximum members of clear factor interactions for \(2^{m-p}\) designs of resolution III and IV. Can. J. Stat. 30, No.~1, 127--136 (2002; Zbl 0999.62059)] is given.