How to implement the creation of an orthogonal array?

Question

Mathematics

An orthogonal array is a two-dimensional array, with such an interesting property - select any 2 columns of the array and find in them all combinations of the values ​​of these columns.
As an example:
A matrix whose columns can be either 1 or 2 contains all possible combinations of three digits:
| 1 | 1 | 1 |
| 2 | 1 | 1 |
| 1 | 2 | 1 |
| 1 | 1 | 2 |
| 2 | 2 | 1 |
| 1 | 2 | 2 |
| 2 | 1 | 2 |
| 2 | 2 | 2 |
Orthogonal array:
| 1 | 1 | 1 |
| 1 | 2 | 2 |
| 2 | 1 | 2 |
| 2 | 2 | 1 |
Can you please tell me a resource \ article \ textbook, preferably in Russian, which describes the compilation of an orthogonal array based on any other?
UPD:
The array definition is taken from the Pairwise testing article. Part 1 - Orthogonal Arrays

Answer 1

As far as I understand, such arrays are described here: habrahabr.ru/post/187882 True, a special case is considered there - when all combinations are required for each pair of columns, and not just those present in another array. But that doesn't make the task any easier.
And orthogonality (in the sense of linear algebra) has nothing to do with it at all.

Answer 2

in fact, you need orthonormal systems, because in fact this is what it is. Read about orthogonalization , there are links to the description of algorithms.

Answer 3

Beklemishev "Analytic geometry and linear algebra"