论文标题
相互正交频率平方的新表示
A new representation of mutually orthogonal frequency squares
论文作者
论文摘要
$ f（mλ;λ）的相互正交频率平方（MOF）$概括了相互正交拉丁正方形的结构：而不是在每个正方形的每一列中完全出现一次$ m $符号，而是每个正方形的每一列，而是重复编号为$λ\ ge 1 $。此类MOF数量的经典上限为$ \ frac {（mλ-1）^2} {m-1} $。我们引入了$ f（mλ;λ）$的MOF的新表示，作为$ \ {0,1 \} $ arrays的线性组合。我们使用此表示形式给出了经典上限的基本证明，以及一组实现上限的MOF的结构约束。然后，当$ m $均匀而$λ$时，我们使用此表示形式为一组$ f（mλ;λ）$的MOF建立最大标准，而$λ$很奇怪，它简化并扩展了先前的分析[T。 Britz，N.J。Cavenagh，A。Mammoliti，I.M. Wanless，相互正交的二元频率正方形，电子。 J. Combin。，27（＃P3.7），2020，26页]，$ m = 2 $和$λ$是奇数的。
Mutually orthogonal frequency squares (MOFS) of type $F(mλ;λ)$ generalize the structure of mutually orthogonal Latin squares: rather than each of $m$ symbols appearing exactly once in each row and in each column of each square, the repetition number is $λ\ge 1$. A classical upper bound for the number of such MOFS is $\frac{(mλ-1)^2}{m-1}$. We introduce a new representation of MOFS of type $F(mλ;λ)$, 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 $F(mλ;λ)$ when $m$ is even and $λ$ is odd, which simplifies and extends a previous analysis [T. Britz, N.J. Cavenagh, A. Mammoliti, I.M. Wanless, Mutually orthogonal binary frequency squares, Electron. J. Combin., 27(#P3.7), 2020, 26 pages] of the case when $m=2$ and $λ$ is odd.
