## (*u*,Â *u*+*v*)-Construction for Nets

Given a (digital) (*t*_{1}, *m*_{1}, *s*_{1})-net and a (digital) (*t*_{2}, *m*_{2}, *s*_{2})-net in in base *b* with *s*_{1} â‰¤ *s*_{2}, a (digital) (*t*, *m*_{1} + *m*_{2}, *s*_{1} + *s*_{2})-net in base *b* with

*t*= max{

*m*

_{1}+

*t*

_{2},

*m*

_{2}â€“

*m*

_{1}+2

*t*

_{1}â€“ 1}

can be constructed [1, Section 5]. In other words, the new net is an OOA with strength *k* = min{2*k*_{1} +1, *k*_{2}}, with *k*_{i} = *m*_{i} â€“ *t*_{i} denoting the strength of the original nets.

Let A_{1} and A_{2} be the ordered orthogonal arrays defined by the original nets. Without loss of generality assume that both OOAs have depth *T * â‰¥ *k* (otherwise pad one of them with zeros). Then the new net is defined by an OOA with the runs

*â€“*

**u***Ï€*(

*),*

**v***)Â : Â*

**v***âˆˆ A*

**u**_{1},

*âˆˆ A*

**v**_{2}}

where *Ï€* : **F**_{b}^{Ts2}â†’**F**_{b}^{Ts1} is the projection selecting the first *s*_{1} blocks of coordinates. If the original nets are linear with generator matrices **H**_{1} and **H**_{2}, both of them with the same depth *T * â‰¥ *k*, a generator matrix of the new net is given by

with *Ï€*(**H**_{2}) denoting the first *Ts*_{1} columns from **H**_{2}.

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 â€œ(*u*, *v*)-constructionâ€.

### See Also

Generalization for arbitrary OOAs

Corresponding result for orthogonal arrays and codes

ConstructionÂ â€œuâ€ in [1].

### References

[1] | JÃ¼rgen Bierbrauer, Yves Edel, and WolfgangÂ Ch. Schmid. Coding-theoretic constructions for ( t, m, s)-nets and ordered orthogonal arrays.Journal of Combinatorial Designs, 10(6):403â€“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*,Â *u*+*v*)-Construction for Nets.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_NUUPlusV.html