  We define a linear code $C_\eta(\delta_T,\delta_X)$ by evaluating polynomials
of bidegree $(\delta_T,\delta_X)$ in the Cox ring on $\mathbb{F}_q$-rational
points of the Hirzebruch surface of parameter $\eta$ on the finite field
$\mathbb{F}_q$. We give explicit parameters of the code, notably using
Gr\"obner bases. The minimum distance provides an upper bound of the number of
$\mathbb{F}_q$-rational points of a non-filling curve on a Hirzebruch surface.
