(uu+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

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

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

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

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

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. “(uu+v)-Construction for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_OUUPlusV.html

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