Two-dimensional minimax Latin hypercube designs

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the ℓ∞ℓ∞-distance we are able to construct minimax Latin hypercube designs of nn points, and to determine the minimal covering radius, for all nn. For the ℓ1ℓ1-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of nn. We conjecture that the obtained lower bound is attained, except for a few small (known) values of nn. For the ℓ2ℓ2-distance we have generated minimax solutions up to n=27n=27 by an exhaustive search method. The latter Latin hypercube designs are included in the website www.spacefillingdesigns.nl.