论文标题
一个孔相等的HMOL上的下限
A lower bound on HMOLS with equal sized holes
论文作者
论文摘要
众所周知，$ n（n）$，是订单$ n $的相互正交拉丁正方形的最大数量，满足大$ n $的下限$ n（n）\ ge n^{1/14.8} $。对于$ h \ ge 2 $，对于数量$ n（h^n）$，相对较少，它表示“ hmols”或相互正交的拉丁正方形的最大数量，它们具有通用的均值置于固定尺寸$ h $的$ n $孔中。我们概括了一种差异矩阵方法，该方法先前用于HMOL的显式构建体。 R.M.的估计威尔逊（Wilson）的较高的环形数字保证了我们的建筑在适当的有限领域取得了成功。将其馈入通用的产品构建中，我们能够为任何$δ> 2 $和所有$ n> n_0（h，Δ）$建立下限$ n（h^n）\ ge（\ log n）^{1/δ} $。
It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which denotes the maximum number of `HMOLS' or mutually orthogonal latin squares having a common equipartition into $n$ holes of a fixed size $h$. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound $N(h^n) \ge (\log n)^{1/δ}$ for any $δ>2$ and all $n > n_0(h,δ)$.
