(u, u+v)-Construction for OOAs
Given two (linear) ordered orthogonal arrays OOA(Mi, si, Sb, T , ki) with s1 ≤ s2, 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 s1 ≤ s2, 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 s1 ≤ s2. Then the resulting OOA A with parameters OOA(M1M2, s1 + s2, Sb, T , k) is given by
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 s1 ≤ s2. Then the set of vectors
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 = Tsi – ni, 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.
See Also
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)-constructionâ€.
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 |
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. “(u, u+v)-Construction for OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_OUUPlusV.html