$F$-Square and Orthogonal $F$-Squares Design: A Generalization of Latin Square and Orthogonal Latin Squares Design

In this paper we are concerned with a generalization of the concept of Latin squares and orthogonality of Latin squares. The condition that every element appears once in each row and each column is replaced by the condition that it appears the same number of times in each row and each column. We call such squares $F$-squares. The usefulness of introducing and investigating the properties of $F$-squares could be justified in two directions. $F$-squares have meaningful applications in laying out experimental designs as exhibited previously by some authors. Their properties prove to be a useful tool in the studies of existence of orthogonal Latin squares and other combinatorial problems.