Use of primary decomposition of polynomial ideals arising from indicator functions to enumerate orthogonal fractions

A polynomial indicator function of designs is first introduced by Fontana {\it et al}. (2000) for two-level cases. They give the structure of the indicator functions, especially the relation to the orthogonality of designs. These results are generalized by Aoki (2019) for general multi-level cases. As an application of these results, we can enumerate all orthogonal fractional factorial designs with given size and orthogonality using computational algebraic software. For example, Aoki (2019) gives classifications of orthogonal fractions of 2x2x2x2x3 designs with strength 3, which is derived by simple eliminations of variables. However, the computational feasibility of this naive approach depends on the size of the problems. In fact, it is reported that the computation of orthogonal fractions of 2x2x2x2x3 designs with strength 2 fails to carry out in Aoki (2019). In this paper, using the theory of primary decomposition, we enumerate and classify orthogonal fractions of 2x2x2x2x3 designs with strength 2. We show there are 35200 orthogonal half fractions of 2x2x2x2x3 designs with strength 2, classified into 63 equivalent classes.