Several Constructions Of Optimal Locally Repairable Code Based On Parity-check Matrix

Locally repairable codes(LRCs)is a class of erasure codes that are widely used in distributed storage systems.There are many results on the bounds and constructions of LRCs,most of which are based on the generator matrix of linear codes.In 2016,Xia and others presented a new framework of parity-check matrix to ensure locality and minimum distance of LRCs.In addition,Xia et al.gave the connection between binary LRCs and binary low-density parity-check codes(LDPC).Thus binary LRCs can be constructed using binary LDPC codes.This thesis is inspired by the above work.We obtain three optimal LRCs attaining the Singleton-like bound,a binary LDPC code employing the cyclic permutation matrix,a binary single-parity LRC applying the method of spreading codes and a q-ary LRC exploiting the Vandermonde matrix.