Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. Theses subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we construct codes with hierarchical locality from a geometric perspective, using fiber products of curves. We demonstrate how the constructed hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes. This point of view provides natural structures for local recovery at each level in the hierarchy.

翻译：具有局部性的码，也称为局部可恢复码，允许利用其他坐标的适当子集来恢复擦除。这些子集通常具有较小的基数，以便利用有限的网络流量和其他资源进行恢复。层次局部可恢复码允许利用其他符号的集合来恢复擦除，这些集合的大小可根据需要增加，从而恢复更多符号。本文从几何视角出发，利用曲线的纤维积构造具有层次局部性的码。我们展示了所构造的层次码如何被视为Reed-Muller码的穿刺子码。这一视角为层次结构中每一层的局部恢复提供了自然结构。