k = 2 Construction

For all t and b, a construction of a (m−2, m, s)-net in base b with

s = $\displaystyle {\frac{{b^{m}−1}}{{b−1}}}$

is given in [1]. This result is based on the equivalence between (m−2, m, s)-nets in base b and sets of s mutually orthogonal m-dimensional hypercubes of order b [1, Theorem 2]. Even though this method is constructive, it does not necessarily result in a digital net.

In [2, Proposition 2], Niederreiter describes an easy method for constructing a digital (m−2, m, s)-net over Fb with

s = $\displaystyle {\frac{{b^{m}−1}}{{b−1}}}$.

The s 2×m matrices

Ci = $\displaystyle \left(\vphantom{\begin{array}{c} \vec{c}_{1}^{(i)}\\ \vec{c}_{2}^{(i)}\end{array}}\right.$$\displaystyle \begin{array}{c} \vec{c}_{1}^{(i)}\\ \vec{c}_{2}^{(i)}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} \vec{c}_{1}^{(i)}\\ \vec{c}_{2}^{(i)}\end{array}}\right)$

for i = 1,…, s are constructed by choosing s pairwise linearly independent vectors for c1(i) (there are exactly s such vectors, e.g., all non-zero vectors with the leftmost non-zero coefficient equal to one) and setting c2(i) to any vector linearly independent to c1(i) for i = 1,…, s.

The case where t = 0 was already given in [3, Theorem 5.4]. This construction can also be seen as a net-embedding of the dual orthogonal array of a linear [s, s−m, 3]-Hamming code over Fb, which is first embedded in an ordered orthogonal array with depth 2 using adding a kth column and then into a net using net from OOA [4, Section 7].

See Also

References

[1]Gary L. Mullen and G. Whittle.
Point sets with uniformity properties and orthogonal hypercubes.
Monatshefte für Mathematik, 113(4):265–273, December 1992.
doi:10.1007/BF01301071
[2]Harald Niederreiter.
A combinatorial problem for vector spaces over finite fields.
Discrete Mathematics, 96(3):221–228, December 1991.
doi:10.1016/0012-365X(91)90315-S MR1139449 (92j:11150)
[3]Harald Niederreiter.
Point sets and sequences with small discrepancy.
Monatshefte für Mathematik, 104(4):273–337, December 1987.
doi:10.1007/BF01294651 MR918037 (89c:11120)
[4]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
[5]Gary L. Mullen, Arijit Mahalanabis, and Harald Niederreiter.
Tables of (t, m, s)-net and (t, s)-sequence parameters.
In Harald Niederreiter and Peter Jau-Shyong Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106 of Lecture Notes in Statistics, pages 58–86. Springer-Verlag, 1995.

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. “k = 2 Construction.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_NK2.html

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