## Ovoid Cap Is Optimal

For *b* > 2, the ovoid, a cap with *b*^{2} + 1 points in the projective space PG(3, *b*), is the largest cap that can exist in such a space. Therefore, for every linear orthogonal array OA(*b*^{4}, *s*,**F**_{b}, 3) / linear [*s*, *s*−4, 4]-code over **F**_{b} we must have *s* ≤ *b*^{2} + 1.

This result is due to [1] for *b* odd and [2] for *b* even.

### See Also

[3, Theorem 16.54]

### References

[1] | Raj Chandra Bose. Mathematical theory of the symmetrical factorial design. Sankhyā, 8:107–166, 1947.MR0026781 (10,201g) |

[2] | B. Qvist. Some remarks concerning curves of the second degree in a finite plane. Annales Academiæ Scientiarum Fennicæ. Series A1, 134:1–27, 1952. |

[3] | 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. “Ovoid Cap Is Optimal.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CBoundOvoid.html