Deterministic construction methods for uniform designs

Space-filling designs are useful for exploring the relationship between the response and factors, especially when the true model is unknown. The wrap-around $L_2$-discrepancy is an important measure of the uniformity, and has often been used as a type of space-filling criterion. However, most obtained designs are generated through stochastic optimization algorithms, and cannot achieve the lower bound of the discrepancies and are only nearly uniform. Then deterministic construction methods for uniform designs are desired. This paper constructs uniform designs under the wrap-around $L_2$-discrepancy by generator matrices of linear codes. Several requirements on the generator matrices, such as a necessary and sufficient condition for generating uniform designs, are derived. Based on these, two simple deterministic constructions for uniform designs are given. Some examples illustrate the effectiveness of them. Moreover, the resulting designs can be regarded as a generalization of good lattice point sets, and also enjoy good orthogonality.