Cubature rules of prescribed merit

We introduce a criterion for the evaluation of multidimensional quadrature, or cubature, rules for the hypercube: this is the merit of a rule which is closely related to its trigonometric degree and which reduces to the Zaremba figure of merit in the case of a lattice rule. We derive a family of rules Q(k)(s) having dimension s and merit 2(k). These rules seem to be competitive with lattice rules with respect to the merit that can be achieved with a given number of abscissas.