Ovoid

In the projective space PG(3, b) with b > 2, a (b2 + 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 + Z2 + aZW + W2 = 0,

where aFb is such that the polynomial x2 + ax + 1 has no root in Fb. In PG(3, 2r) 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(b4, b2 +1,Fb, 3), the dual code a linear [b2 +1, b2 − 3, 4]-code over Fb.

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 b2 +1 | b4 − 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 [b2 +1, 4, b2b]-code over Fb is obtained. All its non-zero code words have weight either b(b−1) or b2. The code words with weight b2 identify the affine subcaps with b2 points. Therefore a [b2 +1, 1, b2]-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. “Ovoid.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_COvoid.html

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