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 $N(h^n)\ge {(\log n)}^{1/\delta }$ for any $\delta >2$ and all $n>n_0(h,\delta )$.

众所周知 $ñ（ñ）$，n阶相互正交的拉丁方的最大数目满足下界$ñ（ñ）\ge {ñ}^{\mathrm{1个}/14.8}$对于大n。为了$H\ge \mathrm{2个}$，对数量的了解相对较少 $ñ（H^{ñ}）$，它表示在具有固定大小h的n个孔中具有相同等分的'HMOLS'或相互正交的拉丁方的最大数量。我们概括了以前用于HMOLS的显式构造的差异矩阵方法。对RM Wilson的高圈数估计可以确保我们的构建在适当大的有限域中获得成功。将其输入到广义的产品构造中，我们能够确定下界$ñ（H^{ñ}）\ge {（\mathrm{日志}ñ）}^{\mathrm{1个}/\delta }$ 对于任何 $\delta >\mathrm{2个}$ 和所有 $ñ>{ñ}_0（H，\delta ）$。