부산대학교 도서관

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.