Maximally Recoverable Codes with Hierarchical Locality: Constructions and Field-Size Bounds

TL;DR

This paper introduces hierarchical locality in maximally recoverable codes, providing new constructions and bounds on field size, enhancing code robustness and efficiency in data recovery scenarios.

Contribution

It extends the concept of locality to hierarchical structures, offers new constructions for hierarchical MRCs, and establishes field size bounds.

Findings

Constructed hierarchical local MRCs for all parameters

Provided field size bounds smaller than existing constructions

Derived properties related to punctured codes and minimum distance

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. We also give constructions of MRC with hierarchical locality for some parameters, whose field size is smaller than that of known constructions for general parameters. We also derive a field size lower bound on MRC with hierarchical locality.