## Product of Cap in PG with Cap in AG

Let *C*_{1} be an *s*_{1}-cap in the projective space PG(*u*_{1}, *b*). Let *C*_{2} be an *s*_{2}-cap in PG(*u*_{2}, *b*) and *C*_{2}Ê¹ âŠ† *C*_{2} an *s*_{2}Ê¹-cap in the affine space AG(*u*_{2}, *b*). Then it is shown in [1, TheoremÂ 8] that an (*s*_{1}*s*_{2}Ê¹ + *s*_{2} â€“ *s*_{2}Ê¹)-cap in PG(*u*_{1} + *u*_{2}, *b*) can be constructed.

If *C*_{2} is already in AG(*u*_{2}, *b*) (and therefore *C*_{2} = *C*_{2}Ê¹), this construction reduces to the well-known product construction for caps [2], yielding an (*s*_{1}*s*_{2})-cap in PG(*u*_{1} + *u*_{2}, *b*) based on an *s*_{1}-cap in PG(*u*_{1}, *b*) and an *s*_{2}-cap in AG(*u*_{2}, *b*). The more general result in [1] allows to include the extra points in *C*_{2}Â âˆ–Â *C*_{2}Ê¹ to the final cap.

### Construction

Let

*C*

_{1}= {

*a*

_{1},â€¦,

*a*

_{s1}} âŠ‚

**F**

_{b}

^{u1+1},

and

*C*

_{2}= {,â€¦,,,â€¦,}

with *b*_{j} âˆˆ **F**_{b}^{u2} for *j* = 1,â€¦, *s*_{2}. Then the new cap consists of the disjoint union of the following two sets:

*i*â‰¤

*s*

_{1}, 1 â‰¤

*j*â‰¤

*s*

_{2}Ê¹},

and

*s*

_{2}Ê¹ <

*j*â‰¤

*s*

_{2}},

with cardinality *s*_{1}*s*_{2}Ê¹ and *s*_{2} â€“ *s*_{2}Ê¹, respectively.

The parent code / orthogonal array listed by MinT for this propagation rule is the code equivalent to *C*_{1}. The following pairs *C*_{2}Ê¹ âŠ† *C*_{2} are used depending on *u*_{2}:

### The Trivial Cap for Dimension *u*_{2} = 1

For *u*_{2} = 1, the optimal solution is to use the trivial 2-cap in PG(1, *b*), which is also in AG(1, *b*). In this case, the product construction reduces to the well-known doubling of a cap: Given an *s*_{1}-cap in PG(*u*_{1}, *b*), a (2*s*_{1})-cap in PG(*u*_{1} + 1, *b*) can be constructed.

### The Oval and Hyperoval for Dimension *u*_{2} = 2

For *u*_{2} = 2, the oval and hyperoval are the largest caps in AG(2, *b*) as well as PG(2, *b*). Therefore, one can construct an (*s*_{1}*s*_{2})-cap in PG(*u*_{1} + 2, *b*) with *s*_{2} = *b* + 1 if *b* is odd and *s*_{2} = *b* + 2 if *b* is even.

### The Ovoid for Dimension *u*_{2} = 3

For *u*_{2} = 3, the optimal solution is to use the affine part of the ovoid. It is a *b*^{2}-cap in AG(3, *b*) which can be completed to a (*b*^{2} + 1)-cap in PG(3, *b*).

Using these caps as *C*_{2}Ê¹ and *C*_{2}, respectively, a (*b*^{2}*s*_{1} + 1)-cap in PG(*u*_{1} + 3, *b*) can be constructed based on an *s*_{2}-cap *C*_{1} in PG(*u*_{1}, *b*). Thus, we have just recovered Segreâ€™s construction from [3].

### Dimension *u*_{2} = 4

For *u*_{2} = 4 the optimal solution is not known for all *b*. MinT uses the following caps *C*_{2}Ê¹ in AG(4, *b*). In some cases, this cap is already projectively-complete, so *C*_{2} = *C*_{2}Ê¹ and we can use the original construction from [2].

b | s_{2}Ê¹ | s_{2} | Reference |

3 | 20 | same | Doubling the ovoid yields a cap in AG |

4 | 40 | same | See [4] |

5 | 65 | 66 | computer completed cap |

7 | 127 | 132 | computer completed cap |

8 | 208 | same | computer completed cap |

9 | 210 | same | computer completed cap |

11 | 311 | 316 | computer completed cap |

13 | 387 | 388 | computer completed cap |

16 | 628 | 629 | computer completed cap |

32 | 3136 | same | computer completed cap |

### Dimension *u*_{2} = 5

For *u*_{2} = 5 one can use the series of caps in PG(5, *b*). It follows from the proof of [1, TheoremÂ 14] that each of these projective caps of size *s*_{2} = (*b* â€“ 1)(*b* + 1)^{2} + 4(*b* + 1) contains an affine subcap of size *s*_{2}Ê¹ = (*b* â€“ 1)(*b* + 1)^{2} + 2(*b* + 1). For *b* = 9 one can even show that the 840-cap contains a subcap of size *s*_{2}Ê¹ = 830 (private communication with Yves Edel).

For some small base fields better results are known:

b | s_{2}Ê¹ | s_{2} | Reference |

3 | 45 | 56 | Hill cap |

4 | 120 | 126 | Glynn cap |

5 | 176 | 186 | computer completed cap |

7 | 427 | 434 | computer completed cap |

8 | 694 | 695 | computer completed cap |

### Dimension *u*_{2} = 6

For *u*_{2} = 6, MinT uses the class of (*b*^{4} +2*b*^{2})-caps in PG(6, *u*) resulting from this construction based on two ovoid caps. It is shown in the proof for [1, TheoremÂ 12] that these caps always contain an affine sub-cap of size *s*_{2}Ê¹ = *b*^{4} + *b*^{2} âˆ’ 1.

### See Also

[5, TheoremÂ 16.63]

### References

[1] | Yves Edel and JÃ¼rgen Bierbrauer. Recursive constructions for large caps. Bulletin of the Belgian Mathematical Society. Simon Stevin, 6(2):249â€“258, 1999. |

[2] | A.Â C. Mukhopadhyay. Lower bounds on m_{t}(r, s).Journal of Combinatorial Theory, SeriesÂ A, 25(1):1â€“13, July 1978.doi:10.1016/0097-3165(78)90026-2 |

[3] | Beniamino Segre. Le geometrie di Galois. Annali di Matematica Pura ed Applicata, 48(1):1â€“96, December 1959.doi:10.1007/BF02410658 |

[4] | Yves Edel and JÃ¼rgen Bierbrauer. The largest cap in AG(4, 4) and its uniqueness. Designs, Codes and Cryptography, 29(1â€“2):99â€“104, May 2003.doi:10.1023/A:1024144223076 |

[5] | JÃ¼rgen 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 Cap in PG with Cap in AG.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CCapProduct1.html