(u, u+v)-Construction for Nets

Given a (digital) (t1, m1, s1)-net and a (digital) (t2, m2, s2)-net in in base b with s1 ≤ s2, a (digital) (t, m1 + m2, s1 + s2)-net in base b with

t = max{m1 + t2, m2 – m1 +2t1 – 1}

can be constructed [1, Section 5]. In other words, the new net is an OOA with strength k = min{2k1 +1, k2}, with ki = mi – ti denoting the strength of the original nets.

Let A1 and A2 be the ordered orthogonal arrays defined by the original nets. Without loss of generality assume that both OOAs have depth T ≥ k (otherwise pad one of them with zeros). Then the new net is defined by an OOA with the runs

{(u – π(v),v)  :  u ∈ A1,v ∈ A2}

where π : FbTs2→FbTs1 is the projection selecting the first s1 blocks of coordinates. If the original nets are linear with generator matrices H1 and H2, both of them with the same depth T ≥ k, a generator matrix of the new net is given by

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

with π(H2) denoting the first Ts1 columns from H2.

This result is an analog to the (u, u + v)-construction in coding theory. A weaker construction yielding nets of the same size, but with larger t-parameter is the direct product of nets, which can be seen as a “(u, v)-construction”.

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. “(u, u+v)-Construction for Nets.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_NUUPlusV.html

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