Design Enumeration through Projection and Isomorphism Examination Based on Counting Vector

Abstract

Design enumeration and isomorphism examination are two difficult tasks in the study of experimental design. The two tasks are different but often related to each other. Design enumeration is concerned with the complete generation of certain types of designs, such as regular designs or orthogonal arrays. Isomorphism examination, on the other hand, compares whether two designs have the same structure subject to some row and column operations. During the process of design enumeration, it is often required to perform isomorphism examination at some stage to reduce the possible redundant calculation. This dissertation includes two main parts. In the first part, we propose an assembly method based on the indicator function and the projection to enumerate all non-isomorphic two-level designs. The assembly method allows us to generate all designs sequentially in a hierarchical structure. We present an algorithm based on the assembly method and generate a complete catalogue of non-isomorphic designs for some cases. In the second part, we develop an initial screening method based on the counting vector for isomorphism examination. We prove that the method provides a more efficient examination than some methods proposed in other articles. The technique of projection is also applied to improve the examination efficiency.