## Direct Product of Two OOAs

Let A_{1} be a (linear) ordered orthogonal array OOA(*M*_{1}, *s*_{1}, *S*_{b}, *T *, *k*_{1}) and let A_{2} be a (linear) OOA(*M*_{2}, *s*_{2}, *S*_{b}, *T *, *k*_{2}). Then the direct product or direct sum of A_{1} and A_{2} is a (linear) OOA(*M*_{1}*M*_{2}, *s*_{1} + *s*_{2}, *S*_{b}, *T *, *k*) with

*k*= min{

*k*

_{1},

*k*

_{2}}.

Correspondingly, given two (linear) NRT-codes C_{1} and C_{2} with parameters ((*s*_{1}, *T *), *N*_{1}, *d*_{1}) and ((*s*_{1}, *T *), *N*_{2}, *d*_{2}), a new (linear) ((*s*_{1} + *s*_{2}, *T *), *N*_{1}*N*_{2}, *d*)-code with *d* = min{*d*_{1}, *d*_{2}} can be obtained.

The (*u*, *u* + *v*)-construction is always at least as good as the direct product.

### Construction for OOAs

The new OOA is defined by the direct product of the original OOAs. To be more specific, the runs of the new OOA A are given by the multi-set

_{1}⊗A

_{2}:= {(

*,*

**x***) :*

**y***∈ A*

**x**_{1},

*∈ A*

**y**_{2}} ⊆

*S*

_{b}

^{(s1+s2, T)}.

If the original OOAs are linear, the new OOA can be constructed as follows: Let **H**_{i} be the *m*_{i}×(*s*_{i}, *T *) generator matrices of A_{1} and A_{2}. Then a generator matrix of the new OOA is given by

### Construction for Codes

Interestingly, the construction for NRT-codes is identical to the construction for OOAs described above. In other words,

_{1}⊗A

_{2}= (A

_{1}

^{⊥}⊗A

_{2}

^{⊥})

^{⊥}.

### See Also

Special case for OAs / codes

For

*T*→∞ one obtains the corresponding result for nets

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Direct Product of Two OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OProduct.html