LP Bound with Quadratic Polynomials for OOAs

In [1, Theorem 10] Bierbrauer shows that for every ordered orthogonal array OOA(M, s, Sb, T , d−1) the condition

M ≥ bTs/$\displaystyle {\frac{{T(b−1)^{2}b^{2T}d}}{{b^{2T+1}-(2T+1)(b−1)b^{T}−1-(\rho_{T}s-d)(b−1)b^{T}(b^{T}−1)}}}$

with

ρT = T – $\displaystyle \sum_{{i=1}}^{{T}}$$\displaystyle {\frac{{1}}{{b^{i}}}}$ = T – $\displaystyle {\frac{{b^{T}−1}}{{b^{T}(b−1)}}}$

must hold, provided that the denominator is positive. The result is established by constructing an explicit solution to the linear programming bound for OOAs based on quadratic polynomials. For T = 1, this bound was already established in [2, Theorem 18.10].

This bound is always weaker than the generalized Plotkin bound for OOAs, provided that the latter is applicable. However, the former yields strong bounds for parameters that are just outside the allowed parameter range of the latter bound.

References

[1]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
[2]Jürgen Bierbrauer.
Introduction to Coding Theory.
Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004.
MR2079734 (2005f:94001)

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. “LP Bound with Quadratic Polynomials for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OBoundLPQuadratic.html

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