A theory of optimum orthogonal fractions is developed for Fourier regression models using integer lattice designs. These provide alternatives to simple grids (product designs) in the case when specified main effects and interaction terms are required to be analyzed. The challenge is to obtain sample sizes which are polynomial in the dimension rather than exponential. This is achieved for certain models with special algorithms based on both algebraic generation and more direct sequential search.
"Lattice-based D-optimum design for Fourier regression." Ann. Statist. 25 (6) 2313 - 2327, December 1997. https://doi.org/10.1214/aos/1030741074