linguistic:paraphrasing
Orthogonal array representation

If each entry of an n × n Latin square is written as a triple (r, c, s), where r is the row, c is the column, and s is the symbol, we obtain a set of n triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the Latin square

is

where for example the triple (2, 3, 1) means that in row 2 and column 3 there is the symbol 1. Orthogonal arrays are usually written in array form where the triples are the rows, such as:

The definition of a Latin square can be written in terms of orthogonal arrays:

This means that the n ordered pairs (r, c) are all the pairs (i, j) with 1 ≤ i, j ≤ n, once each. The same is true of the ordered pairs (r, s) and the ordered pairs (c, s).

The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.