Orthogonal array

Orthogonal array with N runs, k factors, s symbols and strength t is a set of N k-tuples (called runs) with elements from $\{0,\dots ,s-1\}$ such that for every set of t coordinates every combination of symbols in this coordinates appears equal number of times across the runs. The common notion of such orthogonal array is $OA(N,k,s,t)$. It is easy to see, that N is divisible by $s^{t}$ — number of all possible symbol combinations in the t coordinates. The number ${\frac {N}{s^{t}}}$ is called an index of orthogonal array.

Statistical applications

Statistics is a primary application of orthogonal arrays. Experiments based on orthogonal arrays require less tests and yet provide a lot of info.

Particular cases

Some of mathematical constructions are particular cases of orthogonal arrays. For example, latin squares are $OA(n^{2},3,n,2)$. In order to see this, consider all triples $(i,j,s(i,j))$ where $s(i,j)$ — symbol in i-th row and j-th column in the latic square. Then $n^{2}$ such triples for all $i,j\in {0,\dots ,s-1}$ form an orthogonal array with strength 2: there is a single cell with given coordinates, single cell with given row and symbol in the cell and a single cell with given column and symbol in the cell. Here is a simple example:

Latin square	Orthogonal array
1 2 3 3 1 2 2 3 1	(1,1,1) (1,2,2) (1,3,3) (2,1,3) (2,2,1) (2,3,2) (3,1,2) (3,2,3) (3,3,1)

A set of k orthogonal latin square can be converted to $OA(n^{2},k,n,2)$ in a similar way.

Adamar matrices $4N\times 4N$ give an example of $OA(4N,4N-1,2,2)$. You just need to remove a column with 1's from the matrix:

Adamar matrix	Orthogonal array
1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0	(1,1,1) (0,0,1) (0,1,0) (1,0,0)

Another example is a linear error-correcting codes. All codewords of dual code form a linear orthogonal array with strength $d-1$ where $d$ is a distance of the code. Here is an example:

Code	Dual code	Orthogonal array
000 111	000 011 101 110	(0,0,0) (0,1,1) (1,0,1) (1,1,0)

Main results

The main result in theory of orthogonal arrays is the lower linear programming bound on the number of runs in the orthogonal array.

Generalizations

There is several useful generalizations of orthogonal array. We can allow different factors have a different number of symbols. We can assign unequal probabilities to the symbols and require that each combination of symbols appears in the fraction of runs that equal to the product of probabilities of symbols in the combination. And we can union this generalizations allowing different number of symbols and different symbol probabilities for different factors.