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 $).

翻译：在本工作中，我们引入了具有局部性和可用性的最大可恢复码。我们考虑局部可修复码（LRCs），其中$ t $个符号的某个子集各自属于$ N $个局部修复集，这些修复集在移除这$ t $个符号后两两不相交，每个修复集大小为$ r+δ-1 $，且能够本地纠正$ δ-1 $个擦除。当分别设置$ t=1 $和$ t=δ-1=1 $时，经典的具有$ N $个不相交修复集的LRCs和具有$ N $可用性的LRCs得以恢复。允许$ t > 1 $使我们的码能够在相同局部性和可用性条件下降低存储开销。在此框架下，我们将最大可恢复LRCs（MR-LRCs）定义为在给定局部性和可用性约束下，能够纠正任意全局可纠正擦除模式的码。然后，我们识别出一大类可由此类MR-LRCs纠正的全局擦除模式，并证明当$ t=1 $时，这些模式即为所有可纠正模式。我们基于MSRD码提供了三种显式构造的LRCs（因此对于$ t=1 $情况即为MR-LRCs），每种构造在特定参数范围内达到了最小有限域大小。最后，我们将经典MR-LRCs中关于有限域大小的已知下界推广到我们的框架（适用于任意$ t $值）。