Projective Codes Based on the Barlotti Arcs

Let q denote an odd prime power. Let Q denote the conic section in the projective plane PG(2, q) consisting of the points (X : Y : Z) satisfying the equation Y2 = XZ. Then Q consists of the q + 1 points

Q = {(0 : 0 : 0)}∪{(1 : y : y2)  :  Y ∈ Fq}.

The interior points of Q are exactly those points that are neither on Q nor on any tangent of Q. There are exactly $ \binom{q}{2}$ such points, namely all (x : y : z) ∈ PG(2, q) such that y2−xz is a non-square.

These geometric structures are studied in [1], the resulting projective codes in [2, Section 4]: One point of Q together with all interior points of Q defines a projective [s, 3, d]-code over over Fq with

s = $\displaystyle \binom{q}{2}$ +1        and        d = (q – 1)2/2.

Furthermore, Q together with its interior defines a projective [s, 3, d]-code over Fq with

s = $\displaystyle \binom{q+1}{2}$ +1        and        d = (q2 − 1)/2.

See Also

References

[1]Adriano Barlotti.
Some topics in finite geometrical structures.
Mimeo Series 439, Institute of Statistics, Univ. of North Carolina, 1965.
[2]Jürgen Bierbrauer and T. Aaron Gulliver.
New linear codes over F9.
The Australasian Journal of Combinatorics, 21:131–140, 2000.
MR1758264 (2000m:94032)
[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. “Projective Codes Based on the Barlotti Arcs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CBierbrauerGulliverProjective.html

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