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