On Minimum-Point Second-Order Designs

In this note, we discuss k-factor, second order designs with minimum number of points ½(k + l)(k + 2), in particular, those which are extensions of designs that give minimum generalized variance for k = 2 and 3. The experimental region is the unit cuboid. Minimum point designs of this type are unknown for k ≥ 4, and these designs are the best found to date except for k = 4, where a better design is known. Kiefer has shown that these designs cannot be the best for k ≥ 7, via an existence result but, even here, specific better designs are not known and appear difficult to obtain. We also discuss some difficulties of using, in practice, designs that, are D-optimal (that is give minimum generalized variance when the number of points is not restricted).