## Hyperoval

In the projective plane PG(2, *b*) with *b* > 2, the homogenous equation *Z*^{2} = *XY* defines a conic with *b* + 1 points. No three of these points are collinear, so this conic is a (*b* + 1)-cap in PG(2, *b*) known as oval.

If *b* > 2 is even, the tangents of an oval meet in a unique point, the nucleus (0 : 0 : 1). The oval together with its nucleus is a (*b* + 2)-cap in PG(2, *b*), known as hyperoval. The construction of the oval as well as the hyperoval dates back to [1].

Let **F**_{b} = {*x*_{1},…, *x*_{b}}. The resulting orthogonal array is a linear OA(*b*^{3}, *b* + 1,**F**_{b}, 3) with generator matrix

if *b* is odd, and a linear OA(*b*^{3}, *b* + 2,**F**_{b}, 3) with generator matrix

if *b* is even. The resulting code is a linear [*b* + 1, *b*−2, 4]-code over **F**_{b} if *b* is odd, and a linear [*b* + 2, *b*−1, 4]-code over **F**_{b} if *b* is even.

The oval can also be interpreted directly as a projective code, namely the [*b* + 1, 3, *b*−1]-extended Reed-Solomon code RS(3, *b*).

### Optimality

The codes obtained from ovals and hyperovals meet the Singleton bound with equality, and are therefore MDS-codes. The corresponding orthogonal arrays are OAs with index unity.

Ovals and hyperovals are the largest possible caps in PG(2, *b*). Thus the corresponding OAs have the largest number of factors for a tight OA with strength *k* = 3.

The codes and OAs from ovals can also be obtained using Reed-Solomon codes. The ones from hyperovals are strictly better.

### See Also

For

*b*= 4 the points of the hyperoval form a generator matrix of the HexacodeThe hyperoval in PG(2, 2

^{u}) is the smallest member of a family of (*s*,*r*)-arcs, the Denniston arcs[2, pages 253–255]

Oval at

### References

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

[2] | 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. “Hyperoval.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_COval-hyper.html