(Dual) Plotkin Bound for n = 2

It follows immediately from the (dual) Plotkin bound that an orthogonal array OA(bk+t, k + t + n, Sb, k) or a linear [k + t + n, n, k + 1]-code can only exist if

t ≥ (k + 1)$\displaystyle {\frac{{1–1/b^{n−1}}}{{b−1}}}$n + 1.

For n = 2 this yields

t ≥ (k + 1)$\displaystyle {\frac{{1–1/b}}{{b−1}}}$ − 1 = (k + 1)/b−1,

showing that if k turns towards infinity, t must also turn towards infinity.

Thus, an OA(bk+t, k + t + 2, Sb, k) and a linear [k + t + 2, 2, k + 1]-code with fixed finite t cannot exist for arbitrarily large k.

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. “(Dual) Plotkin Bound for n = 2.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CBoundPlotkinInf.html

Show usage of this method