## 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 *Y*^{2} = *XZ*. Then *Q* consists of the *q* + 1 points

*Q*= {(0 : 0 : 0)}∪{(1 :

*y*:

*y*

^{2}) :

*Y*∈

**F**

_{q}}.

The interior points of *Q* are exactly those points that are neither on *Q* nor on any tangent of *Q*. There are exactly such points, namely all (*x* : *y* : *z*) ∈ PG(2, *q*) such that *y*^{2}−*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 **F**_{q} with

*s*= +1 and

*d*= (

*q*– 1)

^{2}/2.

Furthermore, *Q* together with its interior defines a projective [*s*, 3, *d*]-code over **F**_{q} with

*s*= +1 and

*d*= (

*q*

^{2}− 1)/2.

### See Also

[3, pages 265–266]

### 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 F_{9}.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.”
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