Orthogonal array

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

An 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 , … , s − 1 } {\displaystyle \\{0,\dots ,s-1\\}} such that for every set of t coordinates every combination of symbols in these coordinates appears equal number of times across the runs. Such orthogonal arrays are commonly denoted by  O A ( N , k , s , t ) {\displaystyle OA(N,k,s,t)} . It is easy to see that N is divisible by  s t {\displaystyle s^{t}} — the number of all possible symbol combinations in the t coordinates. The number  N s t {\displaystyle {\tfrac {N}{s^{t}}}} is called the index of the orthogonal array.

## Contents

* 1 Statistical applications
* 2 Particular cases
* 3 Main bounds
* 4 Main constructions
* 5 Generalizations

## Statistical applications[edit]

Orthogonal arrays are often used in statistics. Experiments based on orthogonal arrays require less tests and yet provide a lot of info.

## Particular cases[edit]

Some mathematical constructions are particular cases of orthogonal arrays. For example, a Latin square of size  n × n {\displaystyle n\times n} is an orthogonal array with  N = n 2 {\displaystyle N=n^{2}} runs, k = 3 factors, s = n symbols and strength t = 2; in short,  O A ( n 2 , 3 , n , 2 ) {\displaystyle OA(n^{2},3,n,2)} . In order to see this, consider all triples  ( i , j , a ( i , j ) ) {\displaystyle (i,j,a(i,j))} where  a ( i , j ) {\displaystyle a(i,j)} denotes the symbol in the i-th row and j-th column in the Latin square. Then the  n 2 {\displaystyle n^{2}} triples constructed thus for all  i , j ∈ { 0 , … , n − 1 } {\displaystyle i,j\in \\{0,\dots ,n-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

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| 0 | 1 | 2

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2 | 0 | 1
1 | 2 | 0
| (0,0,0)

(0,1,1)
(0,2,2)
(1,0,2)
(1,1,0)
(1,2,1)
(2,0,1)
(2,1,2)
(2,2,0)

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

Adamar matrices of size  4 N × 4 N {\displaystyle 4N\times 4N} give an example of  O A ( 4 N , 4 N − 1 , 2 , 2 ) {\displaystyle OA(4N,4N-1,2,2)} . You just need to remove a column of 1's from the matrix:

Adamar matrix | Orthogonal array

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| 1 | 1 | 1 | 1

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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 code. All codewords of the dual code form a linear orthogonal array with strength  d − 1 {\displaystyle d-1} where  d {\displaystyle d} is a distance of the code. Here is an example:

Code | Dual code | Orthogonal array

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| 000

111
| 000

011
101
110
| (0,0,0)

(0,1,1)
(1,0,1)
(1,1,0)

## Main bounds[edit]

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

## Main constructions[edit]

## Generalizations[edit]

There is several useful generalizations of the definition 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 given by the product of probabilities of symbols in the combination. And we can combine these generalizations, allowing different number of symbols and different symbol probabilities for different factors.