We study cubature formulas for $d$-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree $l$ in dimension $d$, we need about $d^l/l!$ function values if $d$ is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree.
Submitted for publication.