## Doubling a Cap (and Obtaining an Affine Cap)

Given an *s*-cap in the projective space PG(*u*, *b*), a 2*s*-cap in PG(*u* + 1, *b*) can be constructed. For *b* â‰¥ 3 the resulting cap is even in the affine space AG(*u* + 1, *b*). Correspondingly, an orthogonal array OA(*b*^{m+1}, 2*s*,**F**_{b}, 3) can be constructed from an OA(*b*^{m}, *s*,**F**_{b}, 3), and a linear [2*s*, 2*s*âˆ’*m*âˆ’1, 4]-code can be constructed from a linear [*s*, *s*âˆ’*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

with *a* and *b* denoting two different elements from **F**_{b}. In order to obtain a cap in AG(*u* + 1, *b*), *a* and *b* must be non-zero, which is only possible if *b* â‰¥ 3.

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

If

is interpreted as a cap, the resulting cap is the product of**H***H*with the trivial 2-cap in AG(1,*b*).(

*u*,*u*+*v*)-construction based on the original code and the [*s*,*s*âˆ’1, 2]-parity-check-code

### See Also

[1, TheoremÂ 16.60]

### References

[1] | JÃ¼rgen 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).”
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Version: 2024-09-05.
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