Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

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.