Schematic Array With Maximum Rank

As an important structure in algebra and combinatorial mathematics,association scheme ha been widely used in the fields of combinatorial design theory,coding theory and graph theory.Schematic array is the application of association scheme to the design of experiments.An array is called Schematic if its rows form an association scheme with respect to some classification criterion.The type of classification determines the number of classes which is called the rank of an association scheme.Hamming distance has been extensively used to obtain classes of a Schematic array.However,the rank iiiay be too small under this framework.Considering the potential effect of rank on the ability of describing the relationship between row pairs,the Schematic arrays with a big rank is required.,So,we propose some important properties of Schematic arrays.And the upper bound of the rank is obtained respectively depending o n the parity of the number of elements.Furthermore,when the the number of elements is an even.some necessary conditions for the number of elements are given.Next.we propose two methods for constructing the Schematic array with the maximum rank.When the number of rows is odd prime powers.we define special relations based on regular designs to construct Schematic arrays with the maximum rank.Meanwhile,we also construct Schematic arrays with the maximum rank by using the parallel classes of a resolvable balanced incomplete block design to determine the classes when the number of rows is the power of two.