# orthogonal Latin squares

Given two Latin squares $L_{1}=(A,B,C_{1},f_{1})$ and $L_{2}=(A,B,C_{2},f_{2})$ of the same order $n$, we can combine them coordinate-wise to form a single square, whose cells are ordered pairs of elements from $C_{1}$ and $C_{2}$ respectively. Formally, we can form a function $f:A\times B\to C_{1}\times C_{2}$ given by

 $f(i,j)=(f_{1}(i,j),f_{2}(i,j)).$

This function $f$ says that we have created a new square $A\times B$, whose cell $(i,j)$ contains the ordered pair of values, the first coordinate of which corresponds to the value in cell $(i,j)$ of $L_{1}$, and the second to the value in cell $(i,j)$ of $L_{2}$. We may write the combined square $L_{1}*L_{2}$.

For example,

 $\left(\begin{array}[]{cccc}a&b&c&d\\ c&d&a&b\\ d&c&b&a\\ b&a&d&c\end{array}\right)*\left(\begin{array}[]{cccc}1&2&3&4\\ 4&3&2&1\\ 2&1&4&3\\ 3&4&1&2\end{array}\right)=\left(\begin{array}[]{cccc}(a,1)&(b,2)&(c,3)&(d,4)\\ (c,4)&(d,3)&(a,2)&(b,1)\\ (d,2)&(c,1)&(b,4)&(a,3)\\ (b,3)&(a,4)&(d,1)&(c,2)\end{array}\right)$

In general, the combined square is not a Latin square unless the original two squares are equivalent: $f_{1}(i,j)=f_{1}(k,\ell)$ iff $f_{2}(i,j)=f_{2}(k,\ell)$. Nevertheless, the more interesting aspect of pairing up two Latin squares (of the same order) lies in the function $f$:

Definition. We say that two Latin squares are orthogonal if $f$ is a bijection.

Since there are $n^{2}$ cells in the combined square, and $|C_{1}\times C_{2}|=n^{2}$, the function $f$ is a bijection if it is either one-to-one or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.

Remarks.

• The combined square is usually known as a Graeco-Latin square, originated from statisticians Fischer and Yates.

• It can be shown that if $L_{1},\ldots,L_{m}$ are Latin squares of order $n\geq 3$ such that each pair of them are orthogonal, then $m\leq n-1$. If the equality occurs, then the set of Latin squares are said to be complete.

• (Bose) If $n\geq 3$, then $L_{1},\ldots,L_{m}$ form a complete set of pairwise orthogonal Latin squares of order $n$ iff there exists a finite projective plane of order $n$.

## References

• 1 H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 1963
Title orthogonal Latin squares OrthogonalLatinSquares 2013-03-22 16:04:47 2013-03-22 16:04:47 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 62K10 msc 05B15 mutually orthogonal Latin squares MOLS pairwise orthogonal Latin squares complete set of Latin squares