Explicit Constructions of Optimal $(r,\delta)$-Locally Repairable Codes

Locally repairable codes (LRCs) have recently been widely used in distributed storage systems and the LRCs with $(r,\delta)$-locality ($(r,\delta)$-LRCs) attracted a lot of interest for tolerating multiple erasures. Ge et al. constructed $(r,\delta)$-LRCs with unbounded code length and optimal minimum distance when $\delta+1 \leq d \leq 2\delta$ from the parity-check matrix equipped with the Vandermonde structure, but the block length is limited by the size of $\mathbb{F}_q$. In this paper, we propose a more general construction of $(r,\delta)$-LRCs through the parity-check matrix. Furthermore, with the help of MDS codes, we give three classes of explicit constructions of optimal $(r,\delta)$-LRCs with block length beyond $q$. It turns out that 1) our general construction extends the results of Ge et al. and 2) our explicit constructions yield some optimal $(r,\delta)$-LRCs with new parameters.