## 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 = A1A2 := {(x,y)  :  xA1,yA2} ⊆ 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

.

### Construction for Codes

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

A1A2 = (A1A2).