We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.

翻译：我们利用$\mathbb{A}^3$中具有Artin-Schreier型或Kummer型曲线纤维化的曲面，构造了具有分层结构的局部可恢复码。通过利用纤维化的几何与算术性质（该纤维化通过向某一坐标投影得到），我们推导了所构造码的参数。作为副产品，我们获得了特定曲面族中有理点数量的估计（在一种情形下给出了精确计数）。