Several Explicit Constructions Of Optimal Locally Repairable Code

Locally repairable codes(LRCs)are a family of erasure codes which has been proposed for data storage in distributed storage systems recently.Xia et al indicated a connection between a class of binary single-parity locally repairable codes with multiple repair groups and binary low-density parity-check(LDPC)codes.They provided a method to construct binary LRCs from LDPC codes and two specific constructions of binary LRCs that can achieve Singleton-like bound in 2016.However,the constructions of binary LRCs that can achieve Singleton-like bound remain greatly unknown.According to the existing connection between binary LRCs and LDPC codes,this paper presents three classes of binary single-parity LRCs that can achieve Singleton-like bound,based on the cyclic shifts of trace function over a finite field,and incidence matrix which is defined by the inclusion relation between sets,and the Gallager method,and each repair set of these codes contains a single parity symbol.Although the codes constructed in this paper only achieve the trivial minimum distance,the information rate of the first code attains (?) and the rest codes’ information rate attain (?),which is higher than that of the direct product code and the codes constructed by Xia et al.