Explicit Constructions of Optimal (r,δ)-Locally Repairable Codes
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Locally repairable codes (LRCs) have recently been widely used in distributed storage systems and the LRCs with (r,δ)-locality ((r,δ)-LRCs) attracted a lot of interest for tolerating multiple erasures. Ge et al. constructed (r,δ)-LRCs with unbounded code length and optimal minimum distance when δ+1 ≤ d ≤ 2δ from the parity-check matrix equipped with the Vandermonde structure, but the block length is limited by the size of 𝔽_q. In this paper, we propose a more general construction of (r,δ)-LRCs through the parity-check matrix. Furthermore, with the help of MDS codes, we give three classes of explicit constructions of optimal (r,δ)-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,δ)-LRCs with new parameters.
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