Study Of Binary Locally Repairable Codes Over Binary Erasure Channel

Recently,locally repairable codes(LRCs)have been implemented in Distributed Storage Systems(DSSs)to improve the repair efficiency of the systems.LRCs with locality r are a class of(n,k)erasure codes that can recover any coded symbols from r other coded symbols at most.By using binary locally repairable codes(BLRCs)in DSSs,the coding complexity can be significantly reduced due to the elimination of costly multiplication calculations.At present,the studies on the construction of BLRCs has received extensive attention.A large number of studies have proposed the optimal BLRCs construction methods,such as the construction based on the check matrix,the construction based on the generation matrix,the extension of the local dimension and other methods.Based on the existing studies,this paper proposes a construction of BLRCs based on check matrix.In the decoding of BLRCs,there is less related research.It has been proposed to construct BLRCs using Low Density Parity-Check Codes(LDPCs).However,there has been little studies on combining BLRCs decoding algorithm with LDPCs.Therefore,this paper combines the commonality of the two codes,and uses the decoding algorithm of LDPCs to implement the decoding algorithm of BLRCs over binary erasure channel(BEC).The main innovations and research contents of this paper are as follows: Firstly,this paper proposes a method for constructing a class of BLRCs based on the parity check matrices for d=4 and(r+1)| n.In the existing structures of BLRCs based on the parity check matrices,the structures of the parity check matrices are relatively simple.In order to study the performance of BLRCs based on different structures of check matrices,this paper proposes a better search algorithm for the parity check matrices of BLRCs based on mathematical statistics and analysis.For BLRCs that have the same code parameters and different structures of check matrices,the recovery percentage of specific fault blocks can be used to characterize the performance of BLRCs.By analyzing the simulation results,it has been found that the BLRCs constructed in this paper improves the recovery percentages of specific fault blocks compared with the existing BLRCs for r?{ 4,5 }.Secondly,another innovation of this paper is to propose a distributed decoding algorithm for BLRCs over BEC.Based on the deletion decoding algorithm of BLRCs,a distributed decoding algorithm of BLRCs is further proposed according to the actual requirements of the DSSs.This algorithm can significantly reduce the average number of nodes accessed in the process of recovering a fault node over BEC.This paper designs the distributed decoding algorithm of BLRCs from serial aspect and parallel aspect.Through program simulation,the average number of access nodes and fault recovery rate for each decoding algorithm at the occurrence of specific fault blocks are counted,and the BER and FER under different deletion probability are calculated as well.The simulation results show that,whether serial or parallel,the distributed decoding algorithm can significantly reduce the average number of access nodes needed to recover the failed node compared to the deletion decoding algorithm.