On Describing the Incidence Matrix of a Finite Projective Plane Via Orthogonal Latin Squares and Via a Digraph Complete Set of Latin Squares

A fast direct method for obtaining the incidence matrix of a finite projective plane of order \( n \) via \( n-1 \) mutually orthogonal \( n \times n \) Latin squares is described. Conversely, \( n-1 \) mutually orthogonal \( n \times n \) Latin squares are directly exhibited from the incidence matrix of a projective plane of order \( n \). A projective plane of order \( n \) can also be described via a digraph complete set of Latin squares, and a new procedure for doing this will also be described.