Geometric isomorphism check for symmetric factorial designs

Two designs are geometrically isomorphic if one design can be obtained from the other by reordering the runs, relabeling the factors and/or reversing the level order of one or more factors. In this paper, some new necessary and sufficient conditions for identifying geometric isomorphism of symmetric designs with prime levels are provided. A new algorithm for checking geometric isomorphism is proposed and a searching result for geometrically non-isomorphic 3-level orthogonal arrays of 18 runs is presented.