Maximally Recoverable Codes with Hierarchical Locality

TL;DR

This paper introduces maximally recoverable codes with hierarchical locality, extending existing concepts to allow for multi-level locality, and provides constructions for these codes with various parameters and field size optimizations.

Contribution

It defines hierarchical local maximally recoverable codes and presents new constructions, including for cases with a single global parity and lower field sizes.

Findings

Derived properties of punctured codes and minimum distance.

Provided a construction method from hierarchical local to data-local MRCs.

Achieved constructions for all parameters with optimized field sizes.

Abstract

Maximally recoverable codes are a class of codes which recover from all potentially recoverable erasure patterns given the locality constraints of the code. In earlier works, these codes have been studied in the context of codes with locality. The notion of locality has been extended to hierarchical locality, which allows for locality to gradually increase in levels with the increase in the number of erasures. We consider the locality constraints imposed by codes with two-level hierarchical locality and define maximally recoverable codes with data-local and local hierarchical locality. We derive certain properties related to their punctured codes and minimum distance. We give a procedure to construct hierarchical data-local MRCs from hierarchical local MRCs. We provide a construction of hierarchical local MRCs for all parameters. For the case of one global parity, we provide a different construction of hierarchical local MRC over a lower field size.