  In the article we construct low-rate non-split toric $q$-ary codes on some
singular surfaces. More precisely, we consider non-split toric cubic and
quartic del Pezzo surfaces, whose singular points are
$\mathbb{F}_{\!q}$-conjugate. Our codes turn out to be BCH ones with
sufficiently large minimum distance $d$. Indeed, we prove that $d - d^*
\geqslant q - \lfloor 2\sqrt{q} \rfloor - 1$, where $d^*$ is the designed
minimum distance. In other words, we significantly improve upon BCH bound. On
the other hand, the defect of the Griesmer bound for the new codes is
$\leqslant \lfloor 2\sqrt{q} \rfloor - 1$, which also seems to be quite good.
It is worth noting that to better estimate $d$ we actively use the theory of
elliptic curves over finite fields.
