Non-Split Toric BCH Codes on Singular del Pezzo Surfaces

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 Fq-conjugate. Our codes turn out to be BCH ones with sufficiently large minimum distance d. Indeed, we prove that d-d* ≥ q-[2.√q] j-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 ≤ [2.√q] j - 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.