Perfect LRCs and k-Optimal LRCs
Weijun Fang, Bin Chen, Shu-Tao Xia, Fang‐Wei Fu · 2020
Linear codes with locality, called locally repairable codes (LRCs), have been applied in distributed storage systems (DSSs) to minimize the number of storage nodes to be downloaded during repairing a failed node. A linear code has locality r if one can recover an erased code symbol by accessing at most r other code symbols. Bounds and constructions of LRCs have been widely investigated in recent years. In this paper, we first propose the definition of perfect LRCs, whose dimension k achieves the Hamming-type bound proposed by Wang et al. (TIT2019). Then we establish important connections of the existence of LRCs with finite geometry and finite fields, and two systematic constructions of perfect LRCs are obtained. Rewriting the Hamming-type bound by the property of integers, we present a new construction of k-optimal LRCs achieving this bound, which have longer code length than the previously known ones.