Locally Maximal Recoverable (LMR) codes are a subclass of locally recoverable codes, capable of recovering all information-theoretically possible erasures in any one of the local sets while keeping the property of \((r,\delta )\) -locality. In this paper, we define LMR codes with local sets of unequal localities and LMR codes with hierarchical locality. LMR codes with unequal localities are particularly advantageous when erasures occur in local sets with smaller localities. In such a case, fewer helper symbols are needed to recover the codeword. These codes can be helpful in distributed data storage systems where a particular part of the data needs to be accessed more frequently. We also present a construction of distance-optimal LMR codes with unequal localities where the number of elements in a local set need not divide the length of the code. This provides more flexibility in selecting the parameters for these codes. Further, we present two constructions of LMR codes with a two-level hierarchy. A comparison of required field sizes for these constructions is also discussed.
