  For $r \geq 1$ an odd integer, we provide a sequence from the function field
$\mathcal{F}_{q, r}$ of the maximal curve over $\mathbb{F}_{q^{2r}}$ defined by
the affine equation $y^q+y=x^{q^r + 1}$. This sequence has high nonlinear
complexity, and this fact comes from the existence of a rational function on
$\mathcal{F}_{q, r}$ with pole divisor of small degree, and support in certain
$q$ rational places.
