Lower Bound on t for Nets with Strength k = 2 and Arbitrarily Large Dimension s

In [1] and [2, Corollary 5] it is shown that an (m−2, m, s)-net in base b can exist only if

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

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 [2, Theorem 2].

The same bound can also be obtained by extracting the embedded orthogonal array, which has to satisfy the Rao bound. It is included in MinT only for historical reasons, because it is always weaker than the Rao bound for nets.

Results for certain special cases have been obtained earlier: For digital (m−2, m, s)-nets over Fb, the same bound follows easily from the fact that no set of more than

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

linearly independent vectors over Fbm can exist [3, Proposition 3]. See also the bound for linear OAs with strength k = 2. For arbitrary nets, but m restricted to m = 2, the result was shown in [4, Corollary 5.11], using the equivalence between (0, 2, s)-nets in base b and s−2 mutually orthogonal Latin squares of order b.

This bound (together with the trivial bound on s, the trivial bound on t, and the basic propagation rules s-reduction, m-reduction, and t-expansion) is the one tabulated in [5].

References

[1]Harald Niederreiter.
Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes.
Discrete Mathematics, 106/107:361–367, September 1992.
doi:10.1016/0012-365X(92)90566-X MR1181933 (94f:11070)
[2]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
[3]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)
[4]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)
[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. “Lower Bound on t for Nets with Strength k = 2 and Arbitrarily Large Dimension s.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_NBoundK2-SInf.html

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