Orthogonal arrays with variable numbers of symbols are shown to be universally optimal as fractional factorial designs. The orthogonality of completely regular Youden hyperrectangles ($F$-hyperrectangles) is defined as a generalization of the orthogonality of Latin squares, Latin hypercubes, and $F$-squares. A set of mutually orthogonal $F$-hyperrectangles is seen to be a special kind of orthogonal array with variable numbers of symbols. Theorems on the existence of complete sets of mutually orthogonal $F$-hyperrectangles are established which unify and generalize earlier results on Latin squares, Latin hypercubes, and $F$-squares.
"Orthogonal Arrays with Variable Numbers of Symbols." Ann. Statist. 8 (2) 447 - 453, March, 1980. https://doi.org/10.1214/aos/1176344964