Doubling a Cap (and Obtaining an Affine Cap)

Given an s-cap in the projective space PG(u, b), a 2s-cap in PG(u + 1, b) can be constructed. For b ≥ 3 the resulting cap is even in the affine space AG(u + 1, b). Correspondingly, an orthogonal array OA(bm+1, 2s,Fb, 3) can be constructed from an OA(bm, s,Fb, 3), and a linear [2s, 2s−m−1, 4]-code can be constructed from a linear [s, s−m, 4]-code.

Construction

Let H be the cap / a generator matrix of the OA / a parity check matrix of the linear code. Then the resulting cap / generator matrix / parity check matrix is given by

$\displaystyle \left(\vphantom{\begin{array}{cc} \vec{a}_{1\times s} & \vec{b}_{1\times s}\\ \vec{H} & \vec{H}\end{array}}\right.$$\displaystyle \begin{array}{cc} \vec{a}_{1\times s} & \vec{b}_{1\times s}\\ \vec{H} & \vec{H}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc} \vec{a}_{1\times s} & \vec{b}_{1\times s}\\ \vec{H} & \vec{H}\end{array}}\right)$,

with a and b denoting two different elements from Fb. In order to obtain a cap in AG(u + 1, b), a and b must be non-zero, which is only possible if b ≥ 3.

This construction can be seen as a special case of two different, more general constructions:

See Also

References

[1]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. “Doubling a Cap (and Obtaining an Affine Cap).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CCapProduct1Trivial-bound.html

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