Maximally recoverable codes with locality and availability

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 $t$).