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A polynomial indicator function of designs is first introduced by Fontana,Pistone and Rogantin (2000) for two-level designs. They give the structure ofthe indicator function of two-level designs, especially from the viewpoints ofthe orthogonality of the designs. Based on these structure, they use theindicator functions to classify all the orthogonal fractional factorial designswith given sizes using computational algebraic software. In this paper,generalizing the results on two-level designs, the structure of the indicatorfunctions for multi-level designs is derived. We give a system of algebraicequations for the coefficients of indicator functions of fractional factorialdesigns with given orthogonality. We also give another representation of theindicator function, a contrast representation, which reflects the size and theorthogonality of the corresponding design directly. The contrast representationis determined by a contrast matrix, and does not depend on the level-coding,which is one of the advantages of it. We use these results to classifyorthogonal $2^3\times 3$ designs with strength $2$ and orthogonal $2^4\times 3$designs with strength $3$ by computational algebraic software.

Characterizations of indicator functions and contrast representations of fractional factorial designs with multi-level factors