Klein Quartic

The projective plane curve defined by the equation

X3Y + Y3Z + Z3X = 0

is known as Klein quartic [1].

Over F8, the corresponding algebraic function field F = F8(x, y) defined by

x3y + y3 + x = 0

has genus g(F/F8) = 3 and N(F/F8) = 24 rational places [2, Example VI.3.8].

Non-Rational Places

In [3, Example 4.2] it is shown that F/F8 contains at least one place of degree 2.

The Weierstrass Semigroup and the Riemann-Roch Space

A basis of the Riemann-Roch space L(rP) for r ≥ 0 is given by

{1}∪{xiyj  :  0 ≤ i ≤ 2, j ≥ 1, 2i + 3jr}.

The gap numbers of P are {1, 2, 4} [4, Example 3.21].

Optimality

The Klein quartic is optimal because the number of rational places meets Serre’s improved Hasse-Weil bound.

References

[1]Felix Klein.
Über die Transformation siebenter Ordnung der elliptischen Funktionen.
Mathematische Annalen, 14(3):428–471, September 1879.
doi:10.1007/BF01677143
[2]Henning Stichtenoth.
Algebraic Function Fields and Codes.
Springer-Verlag, 1993.
[3]Harald Niederreiter and Chaoping Xing.
Algebraic curves with many rational points over finite fields of characteristic 2.
In Kálmán Győry, Henryk Iwaniec, and Jerzy Urbanowicz, editors, Number Theory in Progress, pages 359–380. W. de Gruyter, Berlin, 1999.
[4]Tom Høholdt, Jacobus H. van Lint, and Ruud Pellikaan.
Algebraic geometry codes.
In Vera S. Pless and W. Cary Huffman, editors, Handbook of Coding Theory, volume 1, pages 871–961. Elsevier Science, 1998.

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. “Klein Quartic.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_FKlein.html

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