Product of Two Caps with Tangent Hyperplane

Let C1 be an (s1 + 1)-cap in the projective space PG(u1, b) and C2 be an (s2 + 1)-cap in PG(u2, b), such that both of them possess a tangent hyperplane. In other words, there are points x1 ∈ C1 and x2 ∈ C2 such that C1 ∖ {x1} is an s1-cap the affine space AG(u1, b) and C2 ∖ {x2} is an s2-cap in AG(u2, b). Then it is shown in [1, Theorem 10] that an (s1s2 + s1 + s2)-cap in PG(u1 + u2, b) can be constructed.

The most common application of this method is to use ovoids as C1 as well as C2. The resulting (b4 +2b2)-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

C1 = {$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ a_{1}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ a_{1}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ a_{1}\end{array}}\right)$,…,$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ a_{s_{1}}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ a_{s_{1}}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ a_{s_{1}}\end{array}}\right)$,$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ a_{\infty}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ a_{\infty}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ a_{\infty}\end{array}}\right)$}

with ai, a∞ ∈ Fbu1, and

C2 = {$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ b_{1}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ b_{1}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ b_{1}\end{array}}\right)$,…,$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ b_{s_{2}}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ b_{s_{2}}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ b_{s_{2}}\end{array}}\right)$,$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ b_{\infty}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ b_{\infty}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ b_{\infty}\end{array}}\right)$}

with bj, b∞ ∈ Fbu2. Then the new cap consists of the disjoint union of the following three sets:

{$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ a_{i}\\ b_{j}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ a_{i}\\ b_{j}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ a_{i}\\ b_{j}\end{array}}\right)$ | 1 ≤ i ≤ s1, 1 ≤ j ≤ s2},
{$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ a_{i}\\ b_{\infty}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ a_{i}\\ b_{\infty}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ a_{i}\\ b_{\infty}\end{array}}\right)$ | 1 ≤ i ≤ s1},

and

{$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ a_{\infty}\\ b_{j}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ a_{\infty}\\ b_{j}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ a_{\infty}\\ b_{j}\end{array}}\right)$ | 1 ≤ j ≤ s2},

with cardinality s1s2, s1, and s2, respectively.

See Also

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.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CCapsAGProduct.html

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