论文标题
混合类型的相互正交二元频率平方
Mutually orthogonal binary frequency squares of mixed type
论文作者
论文摘要
\ emph {频率平方}是一个矩阵，其中每行和列是相同多符号的置换。两个频率平方$ f_1 $和$ f_2 $带有符号多的$ m_1 $和$ m_2 $的\ emph {orthoconal}，如果通过叠加$ f_1 $和$ f_2 $是$ $ m_1 \ $ m_1 \ times m_2 $获得的多组对。一组MOF是一组频率，每对都是正交的。我们首先将经典结合在一组MOF的基数上概括，以覆盖\ emph {混合类型}的情况，这意味着允许符号多组在集合中的正方形之间变化。 如果仅使用符号0和1。我们说，MOFS的$ \ nathcal {f} $是\ emph {type-maximal}，如果不能将其扩展到较大的MOF来通过添加一个符号的符号至少匹配一个$ \ maths $ \ \ \ \ \ \ \ \ \ \ f ima，则频率平方是\ emph {binary}。在Stinson的开创性工作的基础上，最近的几篇论文发现了足以表明一组二元MOF的条件。我们在几个方向上概括了这些论文，发现了暗示类型 - 最大性的新条件。我们的结果涵盖了混合类型的二元频率平方。同样，在以前的论文使用奇偶校验论点的情况下，我们展示了使用模量大于2的论点的优点。
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 $F_1$ and $F_2$ with symbol multisets $M_1$ and $M_2$ are \emph{orthogonal} if the multiset of pairs obtained by superimposing $F_1$ and $F_2$ is $M_1\times M_2$. 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 $\mathcal{F}$ 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 $\mathcal{F}$. 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.
