Mutually orthogonal binary frequency squares of mixed type
A frequency square is a matrix in which each row and column
is a permutation of the same multiset of symbols. Two frequency
squares F1 and F2 with symbol multisets
M1 and M2 are orthogonal if the multiset
of pairs obtained by superimposing F1 and F2 is
M1 × M2. 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 mixed type, meaning that the symbol multisets are allowed to
vary between the squares in the set.
A frequency square is binary if it only uses the symbols 0 and
1. We say that a set F of MOFS is 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 F. 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.