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

Let b ≥ 3 and C1, C2 be si-caps in the projective space PG(ui, b) for i = 1, 2. Then it is shown in [1, Theorem 6] that a (2s1s2)-cap in PG(u1 + u2 + 1, b) can be constructed by first doubling C1 (which yields a (2s1)-cap in the affine space AG(u1 + 1, b)), and then building the product of this cap with C2.

Construction

Let

C1 = {a1,…, as1} ⊂ Fbu1+1,        C2 = {b1,…, bs2} ⊂ Fbu2+1,

and α, β denote distinct, non-zero elements in Fb. Then the new cap is given by

{$\displaystyle \left(\vphantom{\begin{array}{c} \gamma a_{i}\\ b_{j}\end{array}}\right.$$\displaystyle \begin{array}{c} \gamma a_{i}\\ b_{j}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} \gamma a_{i}\\ b_{j}\end{array}}\right)$  :  1 ≤ i ≤ s1, 1 ≤ j ≤ s2, γ ∈ {α, β}}.

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]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: 2024-09-05. http://mint.sbg.ac.at/desc_CCapProduct2Trivial.html

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