Trivial Lower Bound on t

For any (t, m, s)-net in base b, t is always greater or equal to 0.

This follows immediately from the definition of (t, m, s)-nets.

Nets Meeting this Bound

(t, m, s)-nets meeting this bound are (0, m, s)-nets. They have either m = 0 (e.g. trivial nets), m = 1 (e.g. nets with strength k = 1, or s ≤ b + 1 (e.g., nets derived from (0, s)-sequences).

Nets in base b with m > 1 and s > b + 1 must have t ≥ 1 and therefore cannot meet this bound because of s-reduction, m-reduction, and the non-existence of a (0, 2, b + 2)-net due to the mutually orthogonal hypercube bound.

See Also

• Generalization for arbitrary OOAs

• Corresponding results for orthogonal arrays and linear codes and sequences

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. “Trivial Lower Bound on t.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_NBoundTrivialT.html