论文标题
相互正交频率矩形
Mutually orthogonal frequency rectangles
论文作者
论文摘要
类型Fr $（m，n; q）$的频率矩形是$ m \ times n $矩阵，因此每行尺寸$ q $中的每个符号都会出现$ q $ n/q $ times $ n/q $ times，每列中$ m/q $ times。如果在叠加后，每个可能有序的符号出现相同的次数，则认为同一类型的两个频率矩形被认为是正交的。一组$ k $频率矩形，其中每对都是正交的，称为一组相互正交的频率矩形，用$ k $ -MOFR $（M，N; Q）$表示。我们表明，$ k $ -MOFR $（2,2N; 2）$和正交阵列OA $（2N，K，2,2）$等效。我们还表明，OA $（Mn，K，2,2）$表示存在$ K $ -MOFR $（200M，2N; 2）$。我们构建$（4A-2）$ - MOFR $（4,2A; 2）$，假设存在订单$ 4A $的Hadamard矩阵。 a $ k $ - mofr $（m，n; q）$据说为$ t $ - 正交，如果每个尺寸$ t $的每个子集（叠加时）包含$ q^t $的每一个，可能订购的$ t $ t $ t $ t $ tubles $ mn/q^t $ times。如果每个尺寸$ t $的每个子集线性独立，则一组有限字段$ \ mathbb {f} _q $的向量是独立的。我们描述了一种获取一组$ t $ - 正交$ k $ - mofr $（q^m，q^n，q）$对应于$ t $ - $（\ mathbb {f} _q）^{m+n} $的独立向量的$。我们还讨论了$ t $ - 独立向量的上限和下限，并给出了长度为$ n \ leq 16 $的二进制向量的值表。 频率fr $（n，n; q）$的频率矩形称为频率正方形，一组$ k $相互正交的频率平方用$ k $ - mofs $（n; q）$或$ k $ - mofs $（n）$表示。对于$ p $一个奇数的素数，我们表明存在一组$（p-1）$二进制MOFS $（2p）$，因此以$ p \ geq 19 $的先前已知值改善（Britz等，2020）的下限。
A frequency rectangle of type FR$(m,n;q)$ is an $m \times n$ matrix such that each symbol from a set of size $q$ appears $n/q$ times in each row and $m/q$ times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of $k$ frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by $k$--MOFR$(m,n;q)$. We show that a $k$--MOFR$(2,2n;2)$ and an orthogonal array OA$(2n,k,2,2)$ are equivalent. We also show that an OA$(mn,k,2,2)$ implies the existence of a $k$--MOFR$(2m,2n;2)$. We construct $(4a-2)$--MOFR$(4,2a;2)$ assuming the existence of a Hadamard matrix of order $4a$. A $k$--MOFR$(m,n;q)$ is said to be $t$--orthogonal, if each subset of size $t$, when superimposed, contains each of the $q^t$ possible ordered $t$-tuples of entries exactly $mn/q^t$ times. A set of vectors over a finite field $\mathbb{F}_q$ is said to be $t$-independent if each subset of size $t$ is linearly independent. We describe a method to obtain a set of $t$--orthogonal $k$--MOFR$(q^M, q^N, q)$ corresponding to a set of $t$--independent vectors in $(\mathbb{F}_q)^{M+N}$. We also discuss upper and lower bounds on the set of $t$--independent vectors and give a table of values for binary vectors of length $N \leq 16$. A frequency rectangle of type FR$(n,n;q)$ is called a frequency square and a set of $k$ mutually orthogonal frequency squares is denoted by $k$--MOFS$(n;q)$ or $k$--MOFS$(n)$ when there is no ambiguity about the symbol set. For $p$ an odd prime, we show that there exists a set of $(p-1)$ binary MOFS$(2p)$, hence improving the lower bounds in (Britz et al. 2020) for the previously known values for $p \geq 19 $.
