A new representation of mutually orthogonal frequency squares

Abstract

We introduce a new representation of MOFS of type \( F(m\lambda; \lambda) \), as a linear combination of \( \{0,1\} \) arrays. We use this representation to give an elementary proof of the classical upper bound, together with a structural constraint on a set of MOFS achieving the upper bound. We then use this representation to establish a maximality criterion for a set of MOFS of type \( F(m\lambda; \lambda) \) when \( m \) is even and \( \lambda \) is odd, which simplifies and extends a previous analysis \cite{ref3} of the case when \( m = 2 \) and \( \lambda \) is odd.