In this paper, we propose several constructions of Locally Recoverable Codes from elliptic surfaces. In particular, we are able to obtain codes with availability $t>2$, codes with hierarchical locality and, finally, codes which combine availability and hierarchical locality. Our constructions rely on the properties of the torsion groups of elliptic curves and on the fibered structure of elliptic surfaces. In particular, the geometry of the surface is used to introduce a multi-dimensional setting, allowing for more recovery sets, eventually nested one within another.

翻译：本文提出了基于椭圆曲面的局部可恢复码的若干构造方法。特别地，我们能够获得可用性$t>2$的码、具有分层局部性的码，以及最终将可用性与分层局部性相结合的码。我们的构造依赖于椭圆曲线挠群的性质以及椭圆曲面的纤维化结构。具体而言，利用曲面的几何结构引入了多维设置，从而允许更多的恢复集，这些恢复集最终可呈嵌套关系。