Orthogonal array
In mathematics, in the area of combinatorial designs, an orthogonal array is a "table" (array) whose entries come from a fixed finite set of symbols (typically, {1,2,...,n}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here is a simple example of an orthogonal array with symbol set {1,2} and strength 2:
Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns, namely (1,1), (2,1), (1,2) and (2,2) are all the possible ordered pairs of the two element set and each appears exactly once. The second and third columns would give, (1,1), (2,1), (2,2) and (1,2); again, all possible ordered pairs each appearing once. The same statement would hold had the first and second columns been used. This is thus an orthogonal array of strength two.
Orthogonal arrays generalize the idea of mutually orthogonal latin squares in a tabular form. These arrays have many connections to other combinatorial designs and have applications in the statistical design of experiments, coding theory, cryptography and various types of software testing.
A t-(v,k,λ) orthogonal array (t ≤ k) is a λvt × k array whose entries are chosen from a set X with v points such that in every subset of t columns of the array, every t-tuple of points of X appears in exactly λ rows.
In this formal definition, provision is made for repetition of the t-tuples (λ is the number of repeats) and the number of rows is determined by the other parameters.
In many applications these parameters are given the following names:
An orthogonal array is simple if it does not contain any repeated rows.
An orthogonal array is linear if X is a finite field of order q, Fq (q a prime power) and the rows of the array form a subspace of the vector space (Fq)k.
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