Maximally recoverable codes with locality and availability
For coding theory researchers, this provides a new framework and constructions for locally repairable codes with improved storage efficiency, though the results are incremental extensions of existing MR-LRC theory.
This work introduces maximally recoverable codes with locality and availability, generalizing classical LRCs to allow t>1 symbols to share multiple local repair sets, reducing storage overhead. They characterize correctable erasure patterns, provide explicit constructions for t=1 using MSRD codes, and extend lower bounds on field sizes.
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+δ-1 $ and can correct $ δ-1 $ erasures locally. Classical LRCs with $ N $ disjoint repair sets and LRCs with $ N $-availability are recovered when setting $ t = 1 $ and $ t=δ-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 $).