## Direct Product of Two Nets

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

*t*= max{

*m*

_{1}+

*t*

_{2},

*m*

_{2}+

*t*

_{1}}

can be constructed. In other words, the new net has strength *k* = min{*k*_{1}, *k*_{2}}, with *k*_{1} = *m*_{1} – *t*_{1} and *k*_{2} = *m*_{2} – *t*_{2} denoting the strength of the original nets.

The new net is defined by the direct product of the original nets. To be more specific, let **x**_{1},…,**x**_{bm1} and **y**_{1},…,**y**_{bm2} be the points of the original nets. Then the new net is given by the multi-set

**x**_{i},

**y**_{j}) : 1 ≤

*i*≤

*b*

^{m1}, 1 ≤

*j*≤

*b*

^{m2}}.

If the original nets are digital with generator matrices **H**_{1} and **H**_{2} the new net can be constructed as follows: Without loss of generality assume that both matrices have depth *T * (otherwise one of them can be padded with zeros). Then a generator matrix of the new net is given by

The digital case is due to [1, Theorem 10], the general case to [2, Propagation Rule 4].

The (*u*, *u* + *v*)-construction for nets is always at least as good as the direct product.

### See Also

Generalization for arbitrary OOAs

Corresponding result for orthogonal arrays and codes

Propagation Rule 4 in [3]

### References

[1] | Harald Niederreiter and Chaoping Xing. Nets, ( t, s)-sequences, and algebraic geometry.In Peter Hellekalek and Gerhard Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 267–302. Springer-Verlag, 1998. |

[2] | Harald Niederreiter. Constructions of ( t, m, s)-nets.In Harald Niederreiter and Jerome Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998, pages 70–85. Springer-Verlag, 2000. |

[3] | Harald Niederreiter. Constructions of ( t, m, s)-nets and (t, s)-sequences.Finite Fields and Their Applications, 11(3):578–600, August 2005.doi:10.1016/j.ffa.2005.01.001 MR2158777 (2006c:11090) |

### 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. “Direct Product of Two Nets.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_NProduct.html