## Minimax designs in two dimensions

The covering radius ρ of a design of n points in the square is the maximal distance of a point in the square to its closest design point, or equivalently the minimal ρ such that the circles with radius ρ centered at the design points cover the entire square. A remote site is a point of the square that is at distance ρ of the design. The minimax distance design problem is the problem of finding the designs with minimal covering radius. Nurmela and Östergård (2000) determined designs with small covering radius for up to 30 points.

Van Dam (2008) determined all minimax Latin hypercube designs for n up to 27 (up to isomorphism under the action of the symmetry group of the square). For each design the symmetry group is determined too.

The possible symmetry groups, and the corresponding symmetries are:

• The dihedral group D_2: reflections in both diagonals, and rotation over 180 degrees;
• The cyclic group C_4: rotations over 90, 180, and 270 degrees;
• The dihedral group D_1: reflection in one of the diagonals;
• The cyclic group C_2: rotation over 180 degrees;
• The trivial group 1: no symmetries.

In the given figures, an asterisk "*" denotes a remote site.

Note that Van Dam also determined minimax Latin hypercube designs for the distance-measures L1 and L-inf.

 • Dam, E.R. van (2008). Two-dimensional minimax Latin hypercube designs, Discrete Applied Math. 156 (18), 3483-3493. View article View site of author • Nurmela, K.J., and P.R.J. Östergård (2000). Covering a square with up to 30 equal circles, Research Report A62, Helsinki University of Technology, Laboratory for Theoretical Computer Science, Espoo, Finland. View article