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

Abstract

A polynomial indicator function of designs is first introduced by Fontana et al. (J Stat Plan Inference 87:149–172, 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 (J Stat Plan Inference 203:91–105, 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 \(2^4\times 3\) 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 \(2^4\times 3\) 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 \(2^4\times 3\) designs with strength 2. We show there are 35,200 orthogonal half fractions of \(2^4\times 3\) designs with strength 2, classified into 63 equivalent classes.