Some Constructions of Perfect and k-optimal (r,δ)-LRCs

Locally repairable codes (LRCs) are an important class of codes to minimize the number of nodes contacted during repairing in distributed storage systems (DSSs). LRCs with locality (r,δ) were introduced by Prakash et al. that one can recover at most δ−1 erasures code symbols by accessing up to r other code symbols. In this paper, we first propose the Hamming-type bound of (r,δ)-LRCs by extending the definition of ℒ-space proposed by Wang et al. Then we define perfect and k-optimal (r,δ)-LRCs. And we construct two classes of perfect (r = 2,δ)-LRCs which cover the results in [16] for perfect r-LRCs. Meanwhile, we present a construction of k-optimal LRCs for general parameters based on the parity-check matrix and Vandermonde matrix.