## 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, yFb 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 .