Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These 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 describe a hierarchical recovery structure arising from geometry in Reed-Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

翻译：具有局部性的码，也称为局部可恢复码，允许使用其他坐标的适当子集来恢复删除。这些子集通常基数较小，以利于使用有限的网络流量和其他资源进行恢复。分层局部可恢复码允许使用其他符号的集合来恢复删除，这些集合的大小可根据需要增加，以允许恢复更多符号。在本文中，我们描述了由Reed-Muller码中的几何结构以及来自曲线纤维积的具有可用性的码所产生的分层恢复结构。我们展示了如何将纤维积分层码视为Reed-Muller码的穿孔子码，从而统一了这两种构造。这一视角为在分层结构的每一层级上实现具有可用性的局部恢复提供了自然的结构。