Hyperoval

In the projective plane PG(2, b) with b > 2, the homogenous equation Z2 = 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 Fb = {x1,…, xb}. The resulting orthogonal array is a linear OA(b3, b + 1,Fb, 3) with generator matrix

$\displaystyle \left(\vphantom{\begin{array}{cccc} 1 & \cdots & 1 & 0\\ x_{1}^{2} & \cdots & x_{b}^{2} & 1\\ x_{1} & \cdots & x_{b} & 0\end{array}}\right.$$\displaystyle \begin{array}{cccc} 1 & \cdots & 1 & 0\\ x_{1}^{2} & \cdots & x_{b}^{2} & 1\\ x_{1} & \cdots & x_{b} & 0\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cccc} 1 & \cdots & 1 & 0\\ x_{1}^{2} & \cdots & x_{b}^{2} & 1\\ x_{1} & \cdots & x_{b} & 0\end{array}}\right)$

if b is odd, and a linear OA(b3, b + 2,Fb, 3) with generator matrix

$\displaystyle \left(\vphantom{\begin{array}{ccccc} 1 & \cdots & 1 & 0 & 0\\ x… …cdots & x_{b}^{2} & 1 & 0\\ x_{1} & \cdots & x_{b} & 0 & 1\end{array}}\right.$$\displaystyle \begin{array}{ccccc} 1 & \cdots & 1 & 0 & 0\\ x_{1}^{2} & \cdots & x_{b}^{2} & 1 & 0\\ x_{1} & \cdots & x_{b} & 0 & 1\end{array}$$\displaystyle \left.\vphantom{\begin{array}{ccccc} 1 & \cdots & 1 & 0 & 0\\ x… …cdots & x_{b}^{2} & 1 & 0\\ x_{1} & \cdots & x_{b} & 0 & 1\end{array}}\right)$

if b is even. The resulting code is a linear [b + 1, b−2, 4]-code over Fb if b is odd, and a linear [b + 2, b−1, 4]-code over Fb 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

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: 2024-09-05. http://mint.sbg.ac.at/desc_COval-hyper.html

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