Orthogonal Arrays with Variable Numbers of Symbols

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.