linguistic:paraphrasing, linguistic:translation
Nets

A (geometric) (k,n)-net is a set of n elements called points and a set of kn subsets called lines or blocks each of size n with the property that two distinct lines intersect in at most one point. Moreover, the lines can be partitioned into k parallel classes (no two of its lines meet) each containing n lines.

An (n + 1, n)-net is an affine plane of order n.

A set of k MOLS(n) is equivalent to a (k + 2, n)-net.

To construct a (k + 2, n)-net from k MOLS(n), represent the MOLS as an orthogonal array, OA(k + 2, n) (see above). The ordered pairs of entries in each row of the orthogonal array in the columns labeled r and c, will be considered to be the coordinates of the n points of the net. Each other column (that is, Latin square) will be used to define the lines in a parallel class. The n lines determined by the column labeled L will be denoted by l. The points on l will be those with coordinates corresponding to the rows where the entry in the L column is j. There are two additional parallel classes, corresponding to the r and c columns. The lines r and c consist of the points whose first coordinates are j, or second coordinates are j respectively. This construction is reversible.

For example, the OA(5,4) in the above section can be used to construct a (5,4)-net (an affine plane of order 4). The points on each line are given by (each row below is a parallel class of lines):