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):

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

  2. A (8⋅i⋅12⋅112i−1 +10⋅112i)-cap in PG(6i + 3, 3) for all i ≥ 1.

  3. A (16⋅i⋅12⋅112i−1 +56⋅112i)-cap in PG(6i + 5, 3) for all i ≥ 1.

Construction

The new caps are constructed using the projective caps

  1. B0, B ⊂ PG(1, 3) with | B0| = | B| = 2

  2. B0, B ⊂ PG(3, 3) with | B0| = 8 and | B| = 10

  3. B0, B ⊂ PG(5, 3) with | B0| = 16 and | B| = 56

considered in [1, Section 3]; and affine caps A0, A1, A2 ⊂ AG(6i, 3) with | A0| = i⋅12⋅112i−1 and a = | A1| = | A2| = 112i. Based thereupon [1, Theorem 5] yields the claimed (| B0|| A0| + | B| a)-caps.

The affine caps A0, A1, A2 are obtained using [1, Lemma 10] based on the 12-cap C0 ⊂ AG(6, 3), consisting of the vectors of weight 12; on the 112-caps C1, C2 ⊆ AG(6, 3), which are the two different versions of the doubled Hill cap considered in [1, Section 3]; and the admissible set S = I2(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).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CCapsB3.html

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