Complete Sets Of Pairwise Orthogonal Latin Squares Of Order $9$

Abstract

We define two complete sets $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ of pairwise orthogonal $9\times 9$ Latin squares to be equivalent if and only if ${\mathcal{L}}^{\prime }$ can be obtained from $\mathcal{L}$ by some combination of: (i) applying a permutation $\theta$ to the rows of each of the $8$ squares in $\mathcal{L}$, (ii) applying a permutation $\varphi$ to the columns of each square from $\mathcal{L}$, and (iii) permuting the symbols separately within each square from $\mathcal{L}$.
We use known properties of the projective planes of order $9$ to show that, under this equivalence relation, there are $19$ equivalence classes of complete sets. For each equivalence class, we list the species and transformation sets of the $8$ Latin squares in a complete set. As this information alone is not sufficient for determining the equivalence class of a given complete set, we provide a convenient method for doing this.