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 ℤ₂^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, ℤ₂, 3) and a [s, s−m, 4]-code over ℤ₂ 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 ℤ₂ 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: 2024-09-05. http://mint.sbg.ac.at/desc_CB2K3.html