Question

Denition:
An orthogonal array OA(k, n) on n symbols is an n² x k array such that, in any two columns, each ordered pair of symbols occurs exactly once.
Prove that there exists an OA(k, n) if and only if there exist (k - 2) mutually orthogonal Latin squares of order n.

(combinatorics and design)

Answer 1

A Latin square of order n is an n × n array, with symbols in , such that each row and each column contains each of the symbols in exactly once.

1	2	3
3	1	2
2	3	1

1	3	2
3	2	1
2	1	3

Definition (Orthogonal Latin Squares)

Two Latin squares L₁ and L₂ of order n are said to be orthogonal if for every pair of symbols there exist a unique cell with and

L₁ =

1	2	3
3	1	2
2	3	1

L₂ =

1	3	2
3	2	1
2	1	3

(1,1)	(2,3)	(3,2)
(3,3)	(1,2)	(2,1)
(2,2)	(3,1)	(1,3)

Orthogonal Arrays - Redux

An orthogonal array of strength t, a is a array of v symbols, such that in any t columns of the array every one of the possible v^t t-tuples of symbols occurs in exactly rows.

If all the rows of the OA are distinct we call it simple.

If the rows form a subspace of V the OA is said to be linear.