Product of Projective Cap with (Hyper)oval

Let C1 be an s1-cap in the projective space PG(u1, b). Let C2 be an s2-cap in PG(u2, b) and C2ʹ ⊆ C2 an s2ʹ-cap in the affine space AG(u2, b). Then it is shown in [1, Theorem 8] that an (s1s2ʹ + s2 – s2ʹ)-cap in PG(u1 + u2, b) can be constructed.

If C2 is already in AG(u2, b) (and therefore C2 = C2ʹ), this construction reduces to the well-known product construction for caps [2], yielding an (s1s2)-cap in PG(u1 + u2, b) based on an s1-cap in PG(u1, b) and an s2-cap in AG(u2, b). The more general result in [1] allows to include the extra points in C2 ∖ C2ʹ to the final cap.

Construction

Let

C1 = {a1,…, as1} ⊂ Fbu1+1,

and

C2 = {$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ b_{1}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ b_{1}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ b_{1}\end{array}}\right)$,…,$\displaystyle \left(\vphantom{\begin{array}{c} 1\\ b_{s_{2}ʹ}\end{array}}\right.$$\displaystyle \begin{array}{c} 1\\ b_{s_{2}ʹ}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 1\\ b_{s_{2}ʹ}\end{array}}\right)$,$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ b_{s_{2}ʹ+1}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ b_{s_{2}ʹ+1}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ b_{s_{2}ʹ+1}\end{array}}\right)$,…,$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ b_{s_{2}}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ b_{s_{2}}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ b_{s_{2}}\end{array}}\right)$}

with bj ∈ Fbu2 for j = 1,…, s2. Then the new cap consists of the disjoint union of the following two sets:

{$\displaystyle \left(\vphantom{\begin{array}{c} a_{i}\\ b_{j}\end{array}}\right.$$\displaystyle \begin{array}{c} a_{i}\\ b_{j}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} a_{i}\\ b_{j}\end{array}}\right)$  :  1 ≤ i ≤ s1, 1 ≤ j ≤ s2ʹ},

and

{$\displaystyle \left(\vphantom{\begin{array}{c} 0\\ b_{j}\end{array}}\right.$$\displaystyle \begin{array}{c} 0\\ b_{j}\end{array}$$\displaystyle \left.\vphantom{\begin{array}{c} 0\\ b_{j}\end{array}}\right)$  :  s2ʹ < j ≤ s2},

with cardinality s1s2ʹ and s2 – s2ʹ, respectively.

The parent code / orthogonal array listed by MinT for this propagation rule is the code equivalent to C1. The following pairs C2ʹ ⊆ C2 are used depending on u2:

The Trivial Cap for Dimension u2 = 1

For u2 = 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 s1-cap in PG(u1, b), a (2s1)-cap in PG(u1 + 1, b) can be constructed.

The Oval and Hyperoval for Dimension u2 = 2

For u2 = 2, the oval and hyperoval are the largest caps in AG(2, b) as well as PG(2, b). Therefore, one can construct an (s1s2)-cap in PG(u1 + 2, b) with s2 = b + 1 if b is odd and s2 = b + 2 if b is even.

The Ovoid for Dimension u2 = 3

For u2 = 3, the optimal solution is to use the affine part of the ovoid. It is a b2-cap in AG(3, b) which can be completed to a (b2 + 1)-cap in PG(3, b).

Using these caps as C2ʹ and C2, respectively, a (b2s1 + 1)-cap in PG(u1 + 3, b) can be constructed based on an s2-cap C1 in PG(u1, b). Thus, we have just recovered Segre’s construction from [3].

Dimension u2 = 4

For u2 = 4 the optimal solution is not known for all b. MinT uses the following caps C2ʹ in AG(4, b). In some cases, this cap is already projectively-complete, so C2 = C2ʹ and we can use the original construction from [2].

bs2ʹs2Reference
320sameDoubling the ovoid yields a cap in AG
440sameSee [4]
56566computer completed cap
7127132computer completed cap
8208samecomputer completed cap
9210samecomputer completed cap
11311316computer completed cap
13387388computer completed cap
16628629computer completed cap
323136samecomputer completed cap

Dimension u2 = 5

For u2 = 5 one can use the series of caps in PG(5, b). It follows from the proof of [1, Theorem 14] that each of these projective caps of size s2 = (b – 1)(b + 1)2 + 4(b + 1) contains an affine subcap of size s2ʹ = (b – 1)(b + 1)2 + 2(b + 1). For b = 9 one can even show that the 840-cap contains a subcap of size s2ʹ = 830 (private communication with Yves Edel).

For some small base fields better results are known:

bs2ʹs2Reference
34556Hill cap
4120126Glynn cap
5176186computer completed cap
7427434computer completed cap
8694695computer completed cap

Dimension u2 = 6

For u2 = 6, MinT uses the class of (b4 +2b2)-caps in PG(6, u) resulting from this construction based on two ovoid caps. It is shown in the proof for [1, Theorem 12] that these caps always contain an affine sub-cap of size s2ʹ = b4 + b2 − 1.

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]A. C. Mukhopadhyay.
Lower bounds on mt(r, s).
Journal of Combinatorial Theory, Series A, 25(1):1–13, 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):1–96, December 1959.
doi:10.1007/BF02410658
[4]Yves Edel and Jürgen Bierbrauer.
The largest cap in AG(4, 4) and its uniqueness.
Designs, Codes and Cryptography, 29(1–2):99–104, May 2003.
doi:10.1023/A:1024144223076
[5]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 Projective Cap with (Hyper)oval.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CCapProduct1-oval.html

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