A polynomial indicator function of designs is first introduced by Fontana, Pistone and Rogantin (2000) for two-level designs. They give the structure of the indicator function of two-level designs, especially from the viewpoints of the orthogonality of the designs. Based on these structure, they use the indicator functions to classify all the orthogonal fractional factorial designs with given sizes using computational algebraic software. In this paper, generalizing the results on two-level designs, the structure of the indicator functions for multi-level designs is derived. We give a system of algebraic equations for the coefficients of indicator functions of fractional factorial designs with given orthogonality. We also give another representation of the indicator function, a contrast representation, which reflects the size and the orthogonality of the corresponding design directly. The contrast representation is 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 classify orthogonal $2^3times 3$ designs with strength $2$ and orthogonal $2^4times 3$ designs with strength $3$ by computational algebraic software.
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