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.

具有位置性的代码（也称为本地可恢复代码）允许使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，以使用有限的网络流量和其他资源促进恢复。分层本地可恢复代码允许使用其他符号集进行擦除，这些符号的大小会根据需要增加，以允许恢复更多符号。在本文中，我们描述了一种由 Reed-Muller 码中的几何形状产生的分层恢复结构，以及曲线纤维乘积中具有可用性的代码。我们演示了如何将纤维产品分层代码视为 Reed-Muller 代码的穿孔子代码，从而将这两种结构结合在一起。这种观点为本地恢复提供了自然结构，并在层次结构中的每个级别上都提供了可用性。