Orthogonal Arrays

Definition Orthogonal arrays (OAs) are objects that are most often generated via algebraic arguments. They have a number of applications in applied mathematics, and have often been studied by algebraic mathematicians as objects of interest in their own right. Our treatment will reflect their use as representations of statistical experimental designs. An OA is generally presented as a two-dimensional array, table, or matrix of N rows and k columns. Each entry in the array is one element of a set of s “symbols”, often taken to be {0, 1, 2, ..., s− 1} or {1, 2, 3, ..., s}. From the standpoint of basic definitions, the symbols are not regarded as numerical quantities, so {a, b, c, ...} could work just as well. However, the use of numerals (especially the first set above) is convenient for some construction techniques. The final basic quantity required for defining the array it its strength, a positive integer t ≤ k. The single requirement that an N -by-k array of s symbols must meet to be an OA of strength t is that every subset of t columns (from among the k columns), when considered alone, must contain each of the possible s ordered rows the same number of times. A standard notation often used to reference an OA of N rows, k columns, and s symbols, of strength t is OA(N, k, s, t). For example, the following array is an OA(9, 4, 3, 2):