論文

Indicator functions characterize properties of fractional factorial designs such as size and orthogonality via their coefficients. Solving systems of algebraic equations in these coefficients enables the enumeration of orthogonal designs, including classes of mixed-level designs. Counting functions generalize indicator functions by allowing point replication, thereby providing a formulation for the complete enumeration of orthogonal arrays with replicated points. However, the broader range of values for a counting function substantially increases the computational complexity of the resulting algebraic systems. To address this problem, we propose a sequential method that extends an array one factor at a time and solves a sequence of smaller algebraic systems in terms of the number of variables and the degrees of the polynomials based on the theory of projectivity. This approach decomposes the solution of a large zero-dimensional system of polynomial equations into tractable subproblems. We demonstrate the effectiveness of the proposed method by completely enumerating orthogonal arrays such as OA(24,3123,2) and OA(32,4224,3).