## 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*, *S*_{b}, *T *, *k*) can only exist if

*M*≥

*V*

_{b}

^{(s,T )}(

*k*/2),

where *V*_{b}^{(s,T )}(*r*) denotes the volume of a ball with radius *r* in the NRT-space *S*_{b}^{(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 obtainedFor

*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*.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_OBoundRao-KInf.html