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

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 hierarchical locality. We characterize the set of all erasure patterns which can be corrected by hierarchical data-local and hierarchical local maximally recoverable codes (MRC). We show that by carefully puncturing coordinates of the code, hierarchical local MRC can be reduced to hierarchical data-local MRC. Based on picking elements of finite fields and their extensions, all of which satisfy certain linear independence properties, we provide a generic construction of parity check matrix of hierarchical local MRC for all parameters. By appropriately modifying parity check matrices of local MRCs with limited parities, we also give constructions of hierarchical local MRCs with limited parities. Finally, we also derive a lower bound on the field size of hierarchical local MRCs.