## Hermitian Function Field

For *b* = *q*^{2}, the Hermitian function field is an algebraic function field defined by *F* = **F**_{b}(*x*, *y*) with

*y*

^{q}+

*y*=

*x*

^{q+1},

i.e., the trace of *y* equals the norm of *x*. If *q* is a prime, this is an Artin-Schreier extension of the rational function field. The function field corresponds to the projective plane curve defined by the equation

*ZY*

^{q}+

*Z*

^{q}

*Y*=

*X*

^{q+1}.

By [1, Lemma VI.4.4] we have *g*(*F*/**F**_{b}) = *q*(*q*−1)/2 and *N*(*F*/**F**_{b}) = *q*^{3} + 1 rational points, namely *P*_{∞} := (0 : 1 : 0) and the *q*^{3} points *P*_{x,y} := (*x* : *y* : 1) with *x*, *y* ∈ **F**_{b} satisfying tr(*y*) = *N*(*x*).

### Optimality

The Hermitian Function field is maximal, i.e., the number of rational places meets the upper Hasse-Weil bound with equality.

### The Weierstrass Semigroup and the Riemann-Roch Space

A basis of the Riemann-Roch space of L(*rP*_{∞}) is given by

*x*

^{i}

*y*

^{j}:

*i*,

*j*∈ ℕ

_{0},

*j*<

*q*,

*iq*+

*j*(

*q*+ 1) ≤

*r*}.

The Weierstrass semigroup of *P*_{∞} is generated by *q* and *q* + 1. In particular the gap numbers are {1, 2, 5} for *b* = 9 and {1, 2, 3, 6, 7, 11} for *b* = 16.

### Usage in the Context of Digital Sequences

In [2, Proposition 1] *F* is used for constructing the corresponding Niederreiter-Xing-sequence, namely a (*t*, *q*^{3})-sequence over **F**_{q2} with

*t*≤ .

### See Also

### References

[1] | Henning Stichtenoth.Algebraic Function Fields and Codes.Springer-Verlag, 1993. |

[2] | Harald Niederreiter and Chaoping Xing. Low-discrepancy sequences and global function fields with many rational places. Finite Fields and Their Applications, 2(3):241–273, July 1996.doi:10.1006/ffta.1996.0016 MR1398076 (97h:11080) |

[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. “Hermitian Function Field.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_FHermite.html