Optimal Locally Repairable Codes Based On Regular Packing

In the era of big data,disk failures often occur in many systems and result in data loss.Therefore,distributed storage systems emerged.In distributed storage systems,locally repairable codes enable one failed node to be connected to at most others r((?)k)nodes to recover the missing data.The repair e ciency of the storage system can be improved.Under the premise that each repair set contains exactly one check symbol and there are multiple disjoint repair sets,Rawat et al.constructed three types of optimal locally repairable codes.Cai Han,Cheng Minquan,Fan Cuiling,Tang Xiaohu gave the relations between optimal locally repairable codes and combinatorial structure?packings,they constructed some optimal locally repairable codes with information(r,?;1)_c-locality.Later,Jiang Jing,Cheng Minquan used resolvable designs and packings to solve the existence of optimal locally repairable codes with information(3,?;1)_c-locality.When k?20,k(?-1)(?)0,3(mod 4)and(k,?-1)(?){(14,2),(18,2)},the optimal locally repairable codes with information(4,?;1)_c-locality were given by Jiang Jing,Wang Jinyu.Based on the research of Cai Han,Cheng Minquan,Jiang Jing et al.,we will discuss the existence of optimal[n,k,?]_qlocally repairable codes possessing information(4,?;1)_c-locality when the generation matrix of the code contains only partial check column.This dissertation is organized as follows.In Chapter 1,the basic concepts of locally repairable codes and some related combinatorial concepts are introduced.In Chapter 2,some known results about locally repairable codes are given.In Chapter 3,the complete combinatorial characterizations of optimal locally repairable codes with(4,?;1)_c-locality are given.The existence of(k,R;1)-packing with?k(?-1)/4(?) blocks are discussed.Thus some infinity classes of optimal locally repairable codes with(4,?;1)_c-locality are obtained.