Use of polynomial indicator functions to enumerate fractional factorial designs with given properties is first introduced by Fontana, Pistone and Rogantin (2000) for two-level factors, and generalized by Aoki (2019) for multi-level factors. In this paper, we apply this theory to enumerate cross-array designs. For the experiments of several control factors and noise factors, use of the cross-array designs with direct product structure is widespread as an effective robust strategy in Taguchi method. In this paper, we relax this direct product structure to reduce the size of the designs. We obtain 24-runs cross-array designs without direct product structure with some desirable properties for 6 control factors and 3 noise factors, each with two-levels, instead of 32-runs design that is widely used.