On the linear codes with (r, δ)-locality for distributed storage
Jie Hao, Shu‐Tao Xia, Bin Chen · 2017
Recently linear codes with locality properties have attracted a lot of interest due to their desirable applications in distributed storage systems. An [n, k, d] linear code with (r, δ)-locality can enable the local recovery of a failed node in case of more than one node failures. In this paper, we study the theoretical bounds and constructions of linear codes with (r, δ)-locality for all code symbols. A parity-check matrix approach is employed to present an alternate simple proof of the Singleton-like bound for linear codes with all symbol (r, δ)-locality. A refined Singleton-like bound is given for the case that r | k and r + δ - 1 † n. Base on the new proof technique, we enumerate all the possible two classes of optimal binary linear codes meeting the Singleton-like bound. In other words, except the proposed two classes of optimal binary linear codes, there is no other binary linear codes with minimum distance d = n - k - ([k/r] - 1)(δ- 1) + 1.