<p>As an essential structure in algebra and combinatorial mathematics, association schemes have been widely used in combinatorial designs theory, coding theory, and graph theory. Schematic arrays can be regarded as an application of association schemes to the design of experiments. An array is called schematic if its rows could form an association scheme with respect to some classification criterion. According to different criteria, an array can induce different schematic arrays. Simultaneously, different schematic arrays may have different numbers of classes, which is called the rank of a schematic array. Considering the potential effect of rank on the ability to describe the relationships between row pairs, the schematic array with a large rank is preferred. So, the concept of the upper bound of the rank is proposed in this paper for the first time to help construct a schematic array with the maximum rank. Furthermore, the upper bound is obtained respectively depending on the parity of the number of rows. In addition, two methods for constructing schematic arrays with the maximum rank are also proposed. When the number of rows is an odd prime power, we define a special relationship of rows based on regular designs to construct schematic arrays with the maximum rank. Meanwhile, the parallel classes of resolvable balanced incomplete block designs are used to construct schematic arrays with the maximum rank when the number of rows is the power of two.</p>