Maximin Latin Hypercube Designs in Two Dimensions

The problem of finding a maximin Latin hypercube design in two dimensions can be described as positioning n non-attacking rooks on an n x n chessboard such that the minimal distance between pairs of rooks is maximized. Maximin Latin hypercube designs are important for the approximation and optimization of black box functions. In this paper, general formulas are derived for maximin Latin hypercube designs for general n, when the distance measure is l infinity or l1. Furthermore, for the distance measure l2 we obtain maximin Latin hypercube designs for n less than/equal to 70 and approximate maximin Latin hypercube designs for the values of n. We show the reduction in the maximin distance caused by imposing the Latin hypercube design structure is small. This justifies the use of maximin Latin hypercube designs instead of unrestricted designs