Projective Code from 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
where a ∈ Fb 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, b2−b]-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. “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