MinT - Hermitian Function Field

Hermitian Function Field

For b = q², 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^qY = X^q+1.

By [1, Lemma VI.4.4] we have g(F/F_b) = q(q−1)/2 and N(F/F_b) = q³ + 1 rational points, namely P_∞ := (0 : 1 : 0) and the q³ 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

{xⁱy^j : 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 F_q² with

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

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: 2015-09-03. http://mint.sbg.ac.at/desc_FHermite.html