Hermitian Function Field

For b = q2, the Hermitian function field is an algebraic function field defined by F = Fb(x, y) with

yq + y = xq+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

ZYq + ZqY = Xq+1.

By [1, Lemma VI.4.4] we have g(F/Fb) = q(q−1)/2 and N(F/Fb) = q3 + 1 rational points, namely P∞ := (0 : 1 : 0) and the q3 points Px,y := (x : y : 1) with x, y ∈ Fb 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

{xiyj  :  i, j ∈ ℕ0, 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, q3)-sequence over Fq2 with

t ≤ $\displaystyle {\frac{{q^{2}-q}}{{2}}}$.

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: 2024-09-05. http://mint.sbg.ac.at/desc_FHermite.html

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