论文标题
相互正交频率平方的最大集合
Maximal sets of mutually orthogonal frequency squares
论文作者
论文摘要
频率正方形是一个平方矩阵，其中每个行和列是相同多符号的置换。如果频率平方为$（n;λ）$，则它包含$ n/λ$符号，每个符号都会发生$λ$ times每行，每列$λ$ times。在$λ= n/2 $的情况下，我们将频率称为二进制。一组$ k $ -mofs $（n;λ）$是一组$ k $频率的$（n;λ）$的频率平方，使得当叠加任何两个频率平方时，每个可能的有序对均经常出现。 一组$ k $ -maxmofs $（n;λ）$是一组$ k $ -mofs $（n;λ）$，在任何集合（k+1）$ -MOFS $（n;λ）$中不包含。对于$ n $，令$μ（n）$为最小的$ k $，这样就有一组$ k $ -maxmofs $（n; n/2）$。它在[电子中显示。 J. Combin。 27（3）（2020），p3.7] $μ（n）= 1 $，如果$ n/2 $为奇数，而$μ（n）> 1 $如果$ n/2 $均匀。扩展此结果，我们表明，如果$ n/2 $均匀，则$μ（n）> 2 $。另外，我们表明，每当$ n $被$ k $的特定函数排除时，都不存在任何$ k'-maxmofs $（n; n/2）$的$ k'$ k'\ le k $。特别是，这意味着$ \limsupμ（n）$是无限的。然而，我们可以建造固定基数最大二元MOF的无限族。更一般而言，让$ q = p^u $是主要功率，让$ p^v $是$ n $的$ p $的最高功率。如果$ 0 \ le v-uh <u/2 $ for $ h \ ge1 $，则我们证明存在一组$（q^h-1）^2/（q-1）$ - maxmofs $（n; n/q）$。
A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;λ)$ if it contains $n/λ$ symbols, each of which occurs $λ$ times per row and $λ$ times per column. In the case when $λ=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;λ)$ is a set of $k$ frequency squares of type $(n;λ)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;λ)$ is a set of $k$-MOFS$(n;λ)$ that is not contained in any set of $(k+1)$-MOFS$(n;λ)$. For even $n$, let $μ(n)$ be the smallest $k$ such that there exists a set of $k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7] that $μ(n)=1$ if $n/2$ is odd and $μ(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $μ(n)>2$. Also, we show that whenever $n$ is divisible by a particular function of $k$, there does not exist a set of $k'$-maxMOFS$(n;n/2)$ for any $k'\le k$. In particular, this means that $\limsup μ(n)$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides $n$. If $0\le v-uh<u/2$ for $h\ge1$ then we show that there exists a set of $(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.
