The purpose of this paper is to study optimal designs for the elimination of multi-way heterogeneity. The $C$-matrix for the $n$-way heterogeneity setting when $n > 2$ is derived. It turns out to be a natural extension of the known formulas in the lower dimensional case. It is shown that under some regularity, the search for optimal designs can be reduced to that in a lower-way setting. Youden hyperrectangles are defined as higher dimensional generalizations of balanced block designs and generalized Youden designs. When all the sides are equal, they are called Youden hypercubes. It is shown that a Youden hyperrectangle is $E$-optimal and a Youden hypercube is $A$- and $D$-optimal. The latter is quite interesting since it is not always true in two-way settings.
"Optimal Designs for the Elimination of Multi-Way Heterogeneity." Ann. Statist. 6 (6) 1262 - 1272, November, 1978. https://doi.org/10.1214/aos/1176344372