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# Orthogonal Arrays

An orthogonal array (of strength t, with index λ) OAλ(n,m,t) is an m by λnt array with entries from an alphabet of size n with the property that for any choice of t rows each of the possible nt combinations of symbols occurs precisely λ times.

Here we are interested in the case t = 2, λ = 1, so omit these parameters from the notation, writing OA(n,m).

An orthogonal array OA(n,m) is equivalent to a transversal design TD[m;n]. If we pick two rows (that is, two groups of the transversal design) then the symbols found there can be regarded as row and column indices, and we find a set of m – 2 mutually orthogonal Latin squares. In particular, an orthogonal array OA(n,3) with designated pair of rows is equivalent to a Latin square (and hence exists for all n).

If n is a prime power, then TD[m;n] exists if and only if m is at most n + 1. (The design can be obtained from a projective plane of order n by removing a point and nm + 1 lines on that point.)

The block graph of a transversal design TD[m;n] is the graph with the transversals (blocks of size m) as vertices, where blocks are adjacent when they have nonempty intersection. Such a graph is strongly regular, with parameters v = n2, k = m(n – 1), λ = (m – 1)(m – 2) + n – 2, μ = m(m – 1).

For m = 2 this is just the Hamming graph H(2,n), that is, the n×n grid.