## Product of Two Caps with Tangent Hyperplane

Let *C*_{1} be an (*s*_{1} + 1)-cap in the projective space PG(*u*_{1}, *b*) and *C*_{2} be an (*s*_{2} + 1)-cap in PG(*u*_{2}, *b*), such that both of them possess a tangent hyperplane. In other words, there are points *x*_{1} ∈ *C*_{1} and *x*_{2} ∈ *C*_{2} such that *C*_{1} ∖ {*x*_{1}} is an *s*_{1}-cap the affine space AG(*u*_{1}, *b*) and *C*_{2} ∖ {*x*_{2}} is an *s*_{2}-cap in AG(*u*_{2}, *b*). Then it is shown in [1, Theorem 10] that an (*s*_{1}*s*_{2} + *s*_{1} + *s*_{2})-cap in PG(*u*_{1} + *u*_{2}, *b*) can be constructed.

The most common application of this method is to use ovoids as *C*_{1} as well as *C*_{2}. The resulting (*b*^{4} +2*b*^{2})-caps in PG(6, *b*) (cf. [1, Theorem 11] and [2, Table 4.6(i)]) are the largest known caps in the 6-dimensional projective space.

If the resulting cap has a dimension *u* ≠ 6, consult the description of this method in order to find the proper affine caps.

### Construction

Let

*C*

_{1}= {,…,,}

with *a*_{i}, *a*_{∞} ∈ **F**_{b}^{u1}, and

*C*

_{2}= {,…,,}

with *b*_{j}, *b*_{∞} ∈ **F**_{b}^{u2}. Then the new cap consists of the disjoint union of the following three sets:

*i*≤

*s*

_{1}, 1 ≤

*j*≤

*s*

_{2}},

*i*≤

*s*

_{1}},

and

*j*≤

*s*

_{2}},

with cardinality *s*_{1}*s*_{2}, *s*_{1}, and *s*_{2}, respectively.

### See Also

[3, Theorem 16.64]

### 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] | James W. P. Hirschfeld and Leo Storme. The packing problem in statistics, coding theory and finite projective spaces: Update 2001. In Finite Geometries, volume 3 of Developments in Mathematics, pages 201–246. Kluwer Academic Publishers, 2001. |

[3] | 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 Caps with Tangent Hyperplane.”
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Version: 2015-09-03.
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