论文标题
在单数del pezzo表面上的非切片曲面BCH代码
Non-split toric BCH codes on singular del Pezzo surfaces
论文作者
论文摘要
在文章中，我们在某些奇异表面上构建了低速度的非切片折叠$ q $ ary代码。更确切地说，我们考虑了非切片的感谢您的Cutic和Quartic del Pezzo表面，其单数为$ \ Mathbb {f} _ {\！q} $ - conjugate。事实证明，我们的代码是BCH的最小距离$ D $的BCH。确实，我们证明$ d -d^* \ geqslant q - \ lfloor 2 \ sqrt {q} \ rfloor -1 $，其中$ d^* $是设计的最小距离。换句话说，我们对BCH的结合有了显着改善。另一方面，Griesmer绑定的新代码的缺陷为$ \ leqslant \ lfloor 2 \ sqrt {q} \ rfloor -1 $，似乎也很好。值得注意的是，为了更好地估计$ d $，我们积极使用椭圆曲线的理论而不是有限的字段。
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.
