Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2p)+O(3k3)+O(2k)O(2p)+O(3k3)+O(2k) operations to check two 2k−p2k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.