Some Results on Maximally Recoverable Codes with Locality and Hierarchical Locality

Codes with locality allow for efficient recovery from single node failures by minimizing the number of nodes accessed to repair a failed node. These codes can also be extended to handle multiple erasures. Codes with hierarchical locality are another extension of codes with locality, which offer multiple levels of locality as the number of erasures increase. Maximally recoverable codes (MRC) are a class of codes, which satisfy the locality property and in addition also recover from all information theoretically recoverable erasure patterns. In this work, we construct MRC with hierarchical locality based on generator matrices of linearized Reed-Solomon codes and the field size is better than the earlier known construction. We also give a random construction of MRC with hierarchical locality and characterize the field size required. Finally, we present sparse generator matrices for MRC with locality and also sparse and balanced generator matrices for MRC with locality parameter 2 for large set of parameters.