## 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