## Product of Two Projective Caps and an Affine Cap Avoiding Two Hyperplanes

For *b* ≥ 3 let *C*_{i} be an *s*_{i}-cap in the projective space PG(*u*_{i}, *b*) for *i* = 1, 2 and let *C*ʹ be an *s*ʹ-cap in the affine space AG(*u*ʹ, *b*) that avoids not only one, but two hyperplanes. Then it is shown in [1, Corollary 1] (combined with [1, Theorem 3]) that an (*s*_{1}*s*_{2}*s*ʹ)-cap in PG(*u*_{1} + *u*_{2} + *u*ʹ, *b*) can be constructed by building the product of *C*_{1} and *C*ʹ (which yields an (*s*_{1}*s*ʹ)-cap in AG(*u*_{1} + *u*ʹ, *b*)), and then building the product of this cap with *C*_{2}.

The most common choices for *C*ʹ are the trivial 2-cap in AG(1, *b*) and the (hyper)oval in AG(2, *b*). These cases are included in MinT with separate methods, namely here and here, respectively.

In addition to that, one can use the following two computer completed caps from Yves Edel for *C*ʹ: It follows immediately from the weight distribution of the linear orthogonal arrays derived from these caps that the 66-cap in PG(4, 5) contains a 64-cap avoiding two hyperplanes and that the 208-cap in AG(4, 8) also avoids two hyperplanes.

### See Also

[2, Theorem 16.62]

### References

[1] | Yves Edel and Jürgen Bierbrauer. Recursive constructions for large caps. Bulletin of the Belgian Mathematical Society. Simon Stevin, 6(2):249–258, 1999. |

[2] | 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. “Product of Two Projective Caps and an Affine Cap Avoiding Two Hyperplanes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CCapProduct2Other.html