Suzuki Function Field

Let r be a positive integer and let q = 2r and b = 2q2. The Suzuki function field over Fb [1] is an algebraic function field defined by F = Fb(x, y) with

yb + y = xq(xb + x).

It corresponds to the projective plane curve defined by the equation

Zq(Yb + YZb−1) = Xq(Xb + XZb−1).

It has genus g(F/Fb) = q(b−1) and N(F/Fb) = b2 + 1 rational points, namely P := (0 : 1 : 0) and Px,y := (x : y : 1) for all x, yFb.

Optimality

The Suzuki function field is optimal because the number of rational places meets Serre’s improved Hasse-Weil bound.

The Weierstrass Semigroup

The Weierstrass semigroup of P is generated by {b, b + q, b + 2q, b + 2q + 1}. Thus the g = 14 gap numbers for b = 8 are {1, 2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 17, 19, 27}. A basis of the Riemann-Roch space L(rP) can also be found in [1].

Improvements for Suzuki Codes

It is shown in [2] that the true minimum distance of the algebraic-geometric codes derived from the Suzuki curve is d = 56 for r = 10 (i.e., a [64, 3, 56]-code) and d = 42 for r = 23 (i.e., a [64, 11, 42]-code).

See Also

References

[1]Johan P. Hansen and Henning Stichtenoth.
Group codes on certain algebraic curves with many rational points.
Applicable Algebra in Engineering, Communication and Computing, 1(1):67–77, March 1990.
doi:10.1007/BF01810849 MR1325513 (96e:94023)
[2]Chien-Yu Chen and Iwan M. Duursma.
Geometric Reed-Solomon codes of length 64 and 65 over F8.
IEEE Transactions on Information Theory, 49(5):1351–1353, May 2003.
doi:10.1109/TIT.2003.810659 MR1984834
[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. “Suzuki Function Field.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_FSuzuki.html

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