Lattice-based designs with quasi-uniform projections

We provide a theoretical framework to study the projective uniformity of lattice-based designs and give new magic rotation matrices. Utilizing these matrices, we propose an algorithm to generate densest packing-based maximum projection designs, a type of lattice-based designs that process asymptotically optimal separation distance and good projective uniformity. In particular, in two, three, four, six, and eight dimensions, they process asymptotically optimal order of separation and fill distances on every univariate projection, while in four and eight dimensions, they also process asymptotically optimal order of separation distance on multivariate projections. Numerical results imply that these designs are useful when there are likely more than one active variables and more striking if most variables are active.