## Projective Code from Ovoid

In the projective space PG(3, *b*) with *b* > 2, a (*b*^{2} + 1)-cap known as ovoid or ovaloid exists. It can be defined as the set of all points (*W* : *X* : *Y* : *Z*) ∈ PG(3, *b*) such that

*XY*+

*Z*

^{2}+

*aZW*+

*W*

^{2}= 0,

where *a* ∈ **F**_{b} is such that the polynomial *x*^{2} + *ax* + 1 has no root in **F**_{b}. In PG(3, 2^{r}) with *r* ≥ 3, *r* odd, there are also ovoids which are not quadrics, called Tits ovoids (see [1] and [2]).

The resulting orthogonal array is a linear OA(*b*^{4}, *b*^{2} +1,**F**_{b}, 3), the dual code a linear [*b*^{2} +1, *b*^{2} − 3, 4]-code over **F**_{b}.

If *b* is a power of 2, a linear code with the same parameters can also be obtained as narrow-sense BCH-code C({1}) of length *b*^{2} +1 | *b*^{4} − 1. The fact that the minimum distance is at least 4 follows from the Roos-bound and considering the intervals {1, *q*} and { – (*q* + 1), 0} [3].

If the points of an ovoid are interpreted as the columns of the generator matrix of a projective code, a [*b*^{2} +1, 4, *b*^{2}−*b*]-code over **F**_{b} is obtained. All its non-zero code words have weight either *b*(*b*−1) or *b*^{2}. The code words with weight *b*^{2} identify the affine subcaps with *b*^{2} points. Therefore a [*b*^{2} +1, 1, *b*^{2}]-subcode exists, which can be used by construction X.

### Optimality

The ovoid is the largest possible cap in PG(3, *b*).

### See also

### References

[1] | Beniamino Segre. On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two. Acta Arithmetica, 5:315–332, 1959. |

[2] | Jacques Tits. Ovoïdes et groupes de Suzuki. Archiv der Mathemtik, 13(1):187–198, December 1962.doi:10.1007/BF01650065 |

[3] | Chin-Long Chen. Byte-oriented error-correcting codes for semiconductor memory systems. IEEE Transactions on Computers, C−35(7):646–648, July 1986. |

[4] | 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. “Projective Code from Ovoid.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_COvoid-projective.html