Maximally recoverable codes with locality and availability

TL;DR

This paper introduces maximally recoverable codes with locality and availability, expanding classical LRCs to correct more erasure patterns with reduced storage overhead, and provides explicit constructions and bounds.

Contribution

It defines and constructs maximally recoverable LRCs with multiple local repair sets, extending existing theory and providing explicit code constructions with optimal field sizes.

Findings

Identified a large class of correctable global erasure patterns for MR-LRCs.

Provided three explicit constructions of LRCs correcting these patterns, achieving minimal finite-field sizes.

Extended lower bounds on field sizes from classical to the new MR-LRC setting for any t.

Abstract

In this work, we introduce maximally recoverable codes with locality and availability. We consider locally repairable codes (LRCs) where certain subsets of $ t $ symbols belong each to $ N $ local repair sets, which are pairwise disjoint after removing the $ t $ symbols, and which are of size $ r+\delta-1 $ and can correct $ \delta-1 $ erasures locally. Classical LRCs with $ N $ disjoint repair sets and LRCs with $ N $-availability are recovered when setting $ t = 1 $ and $ t=\delta-1=1 $, respectively. Allowing $ t > 1 $ enables our codes to reduce the storage overhead for the same locality and availability. In this setting, we define maximally recoverable LRCs (MR-LRCs) as those that can correct any globally correctable erasure pattern given the locality and availability constraints. We then identify a large class of global erasure patterns that can be corrected by such MR-LRCs and prove that they are all the correctable patterns when $ t=1 $. We provide three explicit constructions of LRCs that can correct such erasure patterns (thus MR-LRCs for $ t=1 $), based on MSRD codes, each attaining the smallest finite-field sizes for some parameter regime. Finally, we extend the known lower bound on finite-field sizes from classical MR-LRCs to our setting (for any value of $ t $).