Revised schematic array

Association scheme was first introduced by statisticians in connection with the design of experiments and has been proven very useful in many fields, including permutation groups, graphs, and coding theory. An array is called schematic if its runs form an association scheme with respect to distance. Schematic arrays, especially schematic orthogonal arrays, have been ideal tools used in designing experiments and generating software test suites. However, the definition of the original schematic array is too demanding. It not only requires the relationship between any two distinct rows, but also overemphasizes the single-row property. This drawback dramatically limits the existence results on schematic arrays. In this article, we modify the original conditions of the association scheme and propose the concept of the revised schematic array. We further elaborate on the rationality of the revised definition. Finally, we also provide two examples of revised schematic arrays, including three-level mirror-symmetric orthogonal arrays and Latin hypercube designs.