Codes with hierarchical locality

In this paper, we study the notion of codes with hierarchical locality that is identified as another approach to local recovery from multiple erasures. The well-known class of codes with locality is said to possess hierarchical locality with a single level. In a code with two-level hierarchical locality, every symbol is protected by an inner-most local code, and another middle-level code of larger dimension containing the local code. We first consider codes with two levels of hierarchical locality, derive an upper bound on the minimum distance, and provide optimal code constructions of low field-size under certain parameter sets. Subsequently, we generalize both the bound and the constructions to hierarchical locality of arbitrary levels.