main diagonal. Form an $(hm+v)\times(hm+v)$ array $A'$ as follows. For $i,j=1,2,\ldots,m$, the subarray in rows $h(i-1)+1, \ldots, hi$ and columns $h(j-1)+1,\ldots,hj$ corresponds to the $(i,j)$-cell in $A$.

We now fill the cells of $A'$. The $h \times h$ subsquares along the diagonal, corresponding to $T_0$, will remain empty. Next, consider the positions arising from the transversals $T_1, \ldots, T_{m-1}$. Partition the last $v$ rows into $m-1$ sets $R_1, R_2,\ldots,R_{m-1}$, with $R_{\ell}$ containing $v_{\ell}$ rows. Similarly partition the last $v$ columns into $C_1, \ldots, C_{m-1}$, where $C_{\ell}$ contains $v_{\ell}$ columns. Suppose that in $A$, cell $(i,j)$, containing entry $\{x_{ij}, y_{ij}, z_{ij}\}$, is in $T_{\ell}$. In the entries of $A'$ arising from this cell, together with the entries in columns $h(j-1)+1,\ldots,hj$ of $R_{\ell}$, the entries in rows $h(i-1)+1,\ldots,hi$ of $C_{\ell}$ and the $v_{\ell} \times v_{\ell}$ subsquare formed by the intersection of $R_{\ell}$ and $C_{\ell}$, place the superimposed three ${\mathrm{IMOLS}}(h+v_{\ell}, v_{\ell})$, so that the missing hole is on the $v_{\ell} \times v_{\ell}$ subsquare formed by the intersection of $R_{\ell}$ and $C_{\ell}$. For these superimposed MOLS, the symbol set is $(\{x_{ij}, y_{ij}, z_{ij}\} \times \mathbb{Z}_h) \cup \{ \infty_{\ell, 1},\infty_{\ell, 2}, \ldots, 
\infty_{\ell, v_{\ell}}\}$, and the missing elements due to the hole are $\infty_{\ell, 1},\infty_{\ell, 