## Generalized (Dual) Plotkin Bound for *n* = 2

In [1, Theorem 2] the Plotkin bound for linear codes is generalized to NRT space. Using duality theory [2] the following bound for linear ordered orthogonal arrays is obtained: For every linear OOA(*b*^{m}, *s*, *S*_{b}, *T *, *k*) we have

*b*

^{Ts−m}– 1) ≤

*b*

^{Ts−m}

*ρ*

_{T }

with

*ρ*

_{T }=

*T*– =

*T*– .

It is easy to see that the original Plotkin bound is obtained for *T * = 1.

In [3, Theorem 5] and [4, Theorem 6] this bound is derived for arbitrary (not necessarily linear) OOAs from the linear programming bound for OOAs.

### Application to Nets and Sequences

In [5, Chapter 3] and [6] Schürer uses the dual Plotkin bound for OOAs for deriving the following bound for (*t*, *s*)-sequences: Let *t*_{b}(*s*) be the minimum *t* such that a (*t*, *s*)-sequence in base *b* can exist. Then

*t*

_{b}(

*s*) ≥

*s*– O(log

*s*)

and therefore

The bound 1/(*b*−1) is the best result for digital as well as arbitrary (*t*, *s*)-sequences known today.

### See Also

For

*T*= 1 the special case for orthogonal arrays and linear codes is obtained.

### References

[1] | Michael Yu. Rosenbloom and Michael A. Tsfasman. Codes for the m-metric.Problems of Information Transmission, 33:55–63, 1997. |

[2] | Harald Niederreiter and Gottlieb Pirsic. Duality for digital nets and its applications. Acta Arithmetica, 97(2):173–182, 2001.MR1824983 (2001m:11130) |

[3] | William J. Martin and Terry I. Visentin. A dual Plotkin bound for ( T , M, S)-nets.IEEE Transactions on Information Theory, 53(1):411–415, January 2007.doi:10.1109/TIT.2006.887514 MR2292900 (2007m:94258) |

[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 |

[5] | Rudolf Schürer.Ordered Orthogonal Arrays and Where to Find Them.PhD thesis, University of Salzburg, Austria, August 2006. |

[6] | Rudolf Schürer. A new lower bound on the t-parameter of (t, s)-sequences.In Alexander Keller, Stefan Heinrich, and Harald Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 623–632. Springer-Verlag, 2008.doi:10.1007/978-3-540-74496-2_37 |

### 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. “Generalized (Dual) Plotkin Bound for *n* = 2.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_OBoundPlotkin-inf.html