## Large Caps in PG(*u*, 3)

The following caps in the projective space PG(*u*, 3) can be constructed based on results in [1] (private communication with Yves Edel):

A (2⋅

*i*⋅12⋅112^{i−1}+2⋅112^{i})-cap in PG(6*i*+ 1, 3) for all*i*≥ 1.A (8⋅

*i*⋅12⋅112^{i−1}+10⋅112^{i})-cap in PG(6*i*+ 3, 3) for all*i*≥ 1.A (16⋅

*i*⋅12⋅112^{i−1}+56⋅112^{i})-cap in PG(6*i*+ 5, 3) for all*i*≥ 1.

### Construction

The new caps are constructed using the projective caps

*B*_{0},*B*⊂ PG(1, 3) with |*B*_{0}| = |*B*| = 2*B*_{0},*B*⊂ PG(3, 3) with |*B*_{0}| = 8 and |*B*| = 10*B*_{0},*B*⊂ PG(5, 3) with |*B*_{0}| = 16 and |*B*| = 56

considered in [1, Section 3]; and affine caps *A*_{0}, *A*_{1}, *A*_{2} ⊂ AG(6*i*, 3) with | *A*_{0}| = *i*⋅12⋅112^{i−1} and *a* = | *A*_{1}| = | *A*_{2}| = 112^{i}. Based thereupon [1, Theorem 5] yields the claimed (| *B*_{0}|| *A*_{0}| + | *B*| *a*)-caps.

The affine caps *A*_{0}, *A*_{1}, *A*_{2} are obtained using [1, Lemma 10] based on the 12-cap *C*_{0} ⊂ AG(6, 3), consisting of the vectors of weight 12; on the 112-caps *C*_{1}, *C*_{2} ⊆ AG(6, 3), which are the two different versions of the doubled Hill cap considered in [1, Section 3]; and the admissible set *S* = *I*_{2}(*i*, 1) defined in [1, Definition 12].

### References

[1] | Yves Edel. Extensions of generalized product caps. Designs, Codes and Cryptography, 31(1):5–14, January 2004.doi:10.1023/A:1027365901231 |

### 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. “Large Caps in PG(*u*, 3).”
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Version: 2008-04-04.
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