Direct Product of Two OOAs

Let A1 be a (linear) ordered orthogonal array OOA(M1, s1, Sb, T , k1) and let A2 be a (linear) OOA(M2, s2, Sb, T , k2). Then the direct product or direct sum of A1 and A2 is a (linear) OOA(M1M2, s1 + s2, Sb, T , k) with

k = min{k1, k2}.

Correspondingly, given two (linear) NRT-codes C1 and C2 with parameters ((s1, T ), N1, d1) and ((s1, T ), N2, d2), a new (linear) ((s1 + s2, T ), N1N2, d)-code with d = min{d1, d2} 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

A = A1⊗A2 := {(x,y)  :  x ∈ A1,y ∈ A2} ⊆ Sb(s1+s2, T).

If the original OOAs are linear, the new OOA can be constructed as follows: Let Hi be the mi×(si, T ) generator matrices of A1 and A2. Then a generator matrix of the new OOA is given by

$\displaystyle \left(\vphantom{\begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times(s_{2},T)}\\ \vec{0}_{m_{2}\times(s_{1},T)} & \vec{H}_{2}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times(s_{2},T)}\\ \vec{0}_{m_{2}\times(s_{1},T)} & \vec{H}_{2}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{H}_{1} & \vec{0}_{m_{1}\times(s_{2},T)}\\ \vec{0}_{m_{2}\times(s_{1},T)} & \vec{H}_{2}\end{array}}\right)$.

Construction for Codes

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

A1⊗A2 = (A1⊥⊗A2⊥)⊥.

See Also

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

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