k = 2 Construction
For all t and b, a construction of a (m−2, m, s)-net in base b with
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
The s 2×m matrices
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
Construction 12 in [5].
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: 2024-09-05.
http://mint.sbg.ac.at/desc_NK2.html