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

Maximally recoverable codes are a class of which recover from all potentially erasure patterns given the locality constraints code. In earlier works, these have been studied in context with locality. The notion has extended to hierarchical locality, allows for gradually increase levels number erasures. We consider imposed by two-level and define maximally characterize set can be corrected data-local local (MRC). show that carefully puncturing coordinates code, MRC reduced MRC. Based on picking elements finite fields their extensions, satisfy certain linear independence properties, we provide generic construction parity check matrix parameters. By appropriately modifying matrices MRCs limited parities, also give constructions parities. Finally, derive lower bound field size MRCs.