## Suzuki Function Field

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

*y*

^{b}+

*y*=

*x*

^{q}(

*x*

^{b}+

*x*).

It corresponds to the projective plane curve defined by the equation

*Z*

^{q}(

*Y*

^{b}+

*YZ*

^{b−1}) =

*X*

^{q}(

*X*

^{b}+

*XZ*

^{b−1}).

It has genus *g*(*F*/**F**_{b}) = *q*(*b*−1) and *N*(*F*/**F**_{b}) = *b*^{2} + 1 rational points, namely *P*_{∞} := (0 : 1 : 0) and *P*_{x,y} := (*x* : *y* : 1) for all *x*, *y* ∈ **F**_{b}.

### 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* + 2*q*, *b* + 2*q* + 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

[3, Section 18.2]

### 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 F_{8}.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