## Product of Two Projective Caps and 2-Cap in AG(1, *b*)

Let *b* ≥ 3 and *C*_{1}, *C*_{2} be *s*_{i}-caps in the projective space PG(*u*_{i}, *b*) for *i* = 1, 2. Then it is shown in [1, Theorem 6] that a (2*s*_{1}*s*_{2})-cap in PG(*u*_{1} + *u*_{2} + 1, *b*) can be constructed by first doubling *C*_{1} (which yields a (2*s*_{1})-cap in the affine space AG(*u*_{1} + 1, *b*)), and then building the product of this cap with *C*_{2}.

### Construction

Let

*C*

_{1}= {

*a*

_{1},…,

*a*

_{s1}} ⊂

**F**

_{b}

^{u1+1},

*C*

_{2}= {

*b*

_{1},…,

*b*

_{s2}} ⊂

**F**

_{b}

^{u2+1},

and *α*, *β* denote distinct, non-zero elements in **F**_{b}. Then the new cap is given by

*i*≤

*s*

_{1}, 1 ≤

*j*≤

*s*

_{2},

*γ*∈ {

*α*,

*β*}}.

### 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 2-Cap in AG(1, *b*).”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CCapProduct2Trivial.html