Existence of schematic arrays under a novel criterion

As an important combinatorial structure, association scheme plays an important role in many fields, such as coding theory, graph theory, and combinatorial design. An array is called a schematic if its runs form an association scheme with respect to a certain classification criterion. However, at present, the existing research results about the schematic array are only based on the distance classification criterion, which dramatically limits the existing results on schematic arrays. This article presents a novel matching classification criterion, called MC-criterion, which, together with the existing distance classification, forms a general framework for classification. An array is called G-schematic if its runs form an association scheme with respect to MC-criterion. Subsequently, this article proves that the existence of two types of G-schematic arrays.