Further Contributions to the Theory of $F$-Squares Design

The main purpose of this paper is four-fold. First, to prove that the best upper bound on the number of mutually orthogonal $F(n, \lambda)$ squares is $(n - 1)^2/(m - 1)$, where $m = n/\lambda$. Second, to show that this upper bound is achievable if $m$ is a prime or prime power and $\lambda = m^h$. Third, to present a set of four mutually orthogonal $F(6; 2)$ squares design. This latter design is important because there are no orthogonal Latin squares of order 6 which could be used for this purpose. Fourth, to indicate a method of composing orthogonal $F$-squares designs. In addition, we have pointed out the way one may construct orthogonal fractional factorial designs and orthogonal arrays from these designs.