Lower Bound on the Number of Runs for OOAs with Strength k = 2 and Arbitrarily Large Number of Factors s
Martin and Stinson [1, Theorem 3.5] show that for an even k ≥ 2 an ordered orthogonal array OOA(M, s, Sb, T , k) can only exist if
where Vb(s,T )(r) denotes the volume of a ball with radius r in the NRT-space Sb(s,T ).
The dual to the Rao bound for OOAs is the Hamming bound or sphere-packing bound in NRT space, which was established in [2, Theorem 3]. Using duality [3], it yields the Rao bound for linear OOAs.
It is shown in [4, Section 5] that the generalized Rao bound follows also from the linear programming bound for OOAs.
See also
For T = 1 the special case for orthogonal arrays and linear codes is obtained
For T →∞ the special case for nets is obtained
References
| [1] | William J. Martin and Douglas R. Stinson. A generalized Rao bound for ordered orthogonal arrays and (t, m, s)-nets. Canadian Mathematical Bulletin, 42(3):359–370, 1999. MR1703696 (2000e:05030) |
| [2] | Michael Yu. Rosenbloom and Michael A. Tsfasman. Codes for the m-metric. Problems of Information Transmission, 33:55–63, 1997. |
| [3] | Harald Niederreiter and Gottlieb Pirsic. Duality for digital nets and its applications. Acta Arithmetica, 97(2):173–182, 2001. MR1824983 (2001m:11130) |
| [4] | Jürgen Bierbrauer. A direct approach to linear programming bounds for codes and tms-nets. Designs, Codes and Cryptography, 42(2):127–143, February 2007. doi:10.1007/s10623-006-9025-6 MR2287187 |
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 the Number of Runs for OOAs with Strength k = 2 and Arbitrarily Large Number of Factors s.”
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