Direct Product of Two Nets

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

t = max{m1 + t2, m2 + t1}

can be constructed. In other words, the new net has strength k = min{k1, k2}, with k1 = m1 – t1 and k2 = m2 – t2 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 x1,…,xbm1 and y1,…,ybm2 be the points of the original nets. Then the new net is given by the multi-set

{(xi,yj)  :  1 ≤ i ≤ bm1, 1 ≤ j ≤ bm2}.

If the original nets are digital with generator matrices H1 and H2 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

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

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

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: 2024-09-05. http://mint.sbg.ac.at/desc_NProduct.html

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