## (u, u+v)-Construction for OOAs

Given two (linear) ordered orthogonal arrays OOA(Mi, si, Sb, T , ki) with s1s2, a (linear) OOA(M1M2, s1 + s2, Sb, T , k) with k = min{2k1 +1, k2} can be constructed [1, Theorem 5.1].

Correspondingly, given two (linear) ((si, T ), Ni, di)-NRT-codes over the same field with s1s2, a (linear) ((s1 + s2, T ), N1N2, d)-code with d = min{2d1, d2} can be constructed [1, Theorem 5.2].

### The Construction for OOAs

Let A1 and A2 denote the OOAs with parameters OOA(M1, s1, Sb, T , k1) and OOA(M2, s2, Sb, T , k2), respectively, and let s1s2. Then the resulting OOA A with parameters OOA(M1M2, s1 + s2, Sb, T , k) is given by

A = {(uπ(v),v)  :  uA1,vA2}

where π : Sb(s2, T)Sb(s1, T) is the projection selecting the first s1 factors.

If A1 and A2 are linear with Mi = bmi and mi×(si, T ) generator matrices Hi, the (m1 + m2)×(s1 + s2, T )generator matrix of A is given by

with π(H2) denoting the first s1 factors of H2.

### The Construction for NRT-Codes

Let C1 be an ((s1, T ), N1, d1)-code and let C2 be an ((s2, T ), N2, d2)-code, both over Fb with s1s2. Then the set of vectors

C = {(u,(u, 0n1×(s2-s1, T)) + v)  :  uC1,vC2}

is an ((s1 + s2, T ), N1N2, d)-code over Fb with d = min{2d1, d2}.

If C1 and C2 are linear with Ni = bni, mi = Tsini, ni×(s, T ) generator matrices Gi, and mi×(si, T ) parity check matrices Hi, the generator matrix of the new linear [(s1 + s2, T ), n1 + n2, d]-code C is given by

,

its parity check matrix is shown above.

### A Special Case

If an OOA(bm2, s2, Sb, T , 3) is combined with the OOA(b, s2, Sb, T , 1) obtained from embedding the parity-check code (which exists for all s ≥ 1), applying the (u, u + v)-construction gives an OOA(bm2+1, 2s, Sb, T , 3), and (after u applications) an OOA(bm2+u, us, Sb, T , 3) all u ≥ 0. This case is treated separately in MinT in order to list a single rule application instead of the full stack.

• Special case for orthogonal arrays / codes

• For T →∞ the corresponding result for nets is obtained

• A weaker construction yielding NRT-codes and OOAs of the same size, but with smaller minimum distance / strength is the direct product, which can be seen as a “(u, v)-construction”.

### References

 [1] Jürgen Bierbrauer, Yves Edel, and Wolfgang Ch. Schmid.Coding-theoretic constructions for (t, m, s)-nets and ordered orthogonal arrays.Journal of Combinatorial Designs, 10(6):403–418, 2002.doi:10.1002/jcd.10015