## Caps in Base *b* = 2

In the projective space PG(*m*−1, 2) over the binary field caps with optimal size are the complement of hyperplanes. They can be constructed easily, e.g. by choosing all *s* = 2^{m−1} vectors in ℤ_{2}^{m} with a fixed (say the first) coordinate equal to 1. Then it is easy to see that any three of these vectors are linearly independent. Therefore a linear orthogonal array OA(2^{m}, *s*, ℤ_{2}, 3) and a [*s*, *s*−*m*, 4]-code over ℤ_{2} with *s* = 2^{m−1} exists for all *m* ≥ 3.

The same codes can also be obtained by taking a [2^{m−1} – 1, 2^{m−1}−*m*, 3]-Hamming code over ℤ_{2} and adding a parity check bit in order to obtain a [2^{m−1}, 2^{m−1}−*m*, 4]-code. Since the Hamming code meets the Hamming bound, the resulting code is also optimal.

### 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. “Caps in Base *b* = 2.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CB2K3.html