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A Generalized Rao Bound for Ordered Orthogonal Arrays and (t, m, s)-Nets

Published online by Cambridge University Press:  20 November 2018

W. J. Martin  and
D. R. Stinson
W. J. Martin
Affiliation:
Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba R3B 2E9
D. R. Stinson
Affiliation:
Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
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Abstract

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In this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and $\left( t,\,m,\,s \right)$-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (i.e., necessary conditions) for existence of these objects.

Keywords

05B1565C99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 3 , 01 September 1999 , pp. 359 - 370
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Clayman, A. T., Lawrence, K. M., Mullen, G. L., Niederreiter, H. and Sloane, N. J. A., Updated tables of parameters of (T, M, S)-nets. J. Combin. Des., to appear (see also http://www.emba.uvm.edu/jcd/toappear.html).Google Scholar
[2] Edel, Y. and Bierbrauer, J., Construction of digital nets from BCH-codes. In: Monte Carlo and Quasi-Monte Carlo Methods 1996 (Salzburg), Lecture Notes in Statist. 127 (1998), 221–231.Google Scholar
[3] Lawrence, K. M., Combinatorial bounds and constructions in the theory of uniform point distributions in unit cubes, connections with orthogonal arrays and a poset generalization of a related problem in coding theory. PhD Thesis, University of Wisconsin-Madison, 1995.Google Scholar
[4] Lawrence, K. M., A combinatorial characterization of (t, m, s)-nets in base b. J. Combin. Des. 4 (1996), 275–293.Google Scholar
[5] Lawrence, K. M., The orthogonal array bound for (t, m, s)-nets is not always attained. Preprint.Google Scholar
[6] Martin, W. J. and Stinson, D. R., Association schemes for ordered orthogonal arrays and (T, M, S)-nets. Canad. J. Math. (2) 51 (1999), 326–346.Google Scholar
[7] Mullen, G. L. and Schmid, W. Ch., An equivalence between (T, M, S)-nets and strongly orthogonal hypercubes. J. Combin. Theory A 76 (1996), 164–174.Google Scholar
[8] Mullen, G. L. and Whittle, G., Point sets with uniformity properties and orthogonal hypercubes. Monatsh. Math. 113 (1992), 265–273.Google Scholar
[9] Niederreiter, H., Point sets and sequences with small discrepancy. Monatsh.Math. 104 (1987), 273–337.Google Scholar
[10] Rao, C. R., Factorial experiments derivable from combinatorial arrangements of arrays. Suppl. J. Roy. Statist. Soc. 9 (1947), 128–139.Google Scholar
[11] Schmid, W. Ch., (T, M, S)-nets: digital constructions and combinatorial aspects. PhD Thesis, Universität Salzburg, 1995.Google Scholar
[12] Schmid, W. Ch. and Wolf, R., Bounds for digital nets and sequences. Acta Arith. 78 (1997), 377–399.Google Scholar
[13] Sobol, M., The distribution of points in a cube and the approximate evaluation of integrals. Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 784–802. In Russian.Google Scholar