Locally Repairable Codes with Heterogeneous Locality Constraints

A code over a finite alphabet is called locally repairable codes (LRCs) if every symbol in the encoding is a function of a small number of other symbols of the codeword. In this paper, we study LRCs with heterogeneous locality constraints. We introduce (n; k; r(i); delta(i); i is an element of [m]) LRCs which generalize the LRCs with equal (r; delta)-locality, and establish the Singleton-like bound for such codes. Then, we study how to construct optimal LRCs, namely, its minimum distance attains the proposed bound. In precisely, we redefine the notation of LRCs with maximal recoverability (MR-LRCs) based on the proposed LRCs and show that MR-LRCs are optimal LRCs. Finally, we construct a family of MR-LRCs which extend the construction of LRCs with equal locality presented by Rawat et al.