Mutually orthogonal binary frequency squares of mixed type
arXiv:2212.01746 · doi:10.1007/978-3-031-48679-1
Abstract
A \emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. Two frequency squares and with symbol multisets and are \emph{orthogonal} if the multiset of pairs obtained by superimposing and is . A set of MOFS is a set of frequency squares in which each pair is orthogonal. We first generalise the classical bound on the cardinality of a set of MOFS to cover the case of \emph{mixed type}, meaning that the symbol multisets are allowed to vary between the squares in the set. A frequency square is \emph{binary} if it only uses the symbols 0 and 1. We say that a set of MOFS is \emph{type-maximal} if it cannot be extended to a larger set of MOFS by adding a square whose symbol multiset matches that of at least one square already in . Building on pioneering work by Stinson, several recent papers have found conditions that are sufficient to show that a set of binary MOFS is type-maximal. We generalise these papers in several directions, finding new conditions that imply type-maximality. Our results cover sets of binary frequency squares of mixed type. Also, where previous papers used parity arguments, we show the merit of arguments that use moduli greater than 2.