| In recent years,with the rapid growth of data resources,storage failures often occur in the storage of data and result in data loss.For this reason,people are looking for solutions that can solve this problem.Compared with the simple straightforward replication,the locally repairable codes can reduce repair cost and improve repair efficiency.They have been widely used in distributed storage system,such as Windows Azure storage system and Facebook.Therefore,locally repairable codes have become a hot topic studied by many scholars.In 2012,Gopalan et al.proposed the concept of repairable locality,and deduced the famous Singleton-like bound for the first time.In 2014,the concept of(r,δ)c-locality was proposed by Wang Anyu et al.In 2016,Rawat et al.gave the upper bound of the minimum distance of[n,k,d]q linear codes with information(r,δ;1)c-locality under the assumption that each repair set contains exactly one check symbol.In 2019,Cai Han et al.gave the equivalent relationship between locally repairable codes and packings under the assumption of Rawat et al.Based on the research of Cai Han et al.,this paper constructs the optimal[n,k,3]q locally repairable codes with information(r,3;1)c-locality.This paper is organized as f’ollows.In Chapter 1,the concepts of linear codes,locally repairable codes and some combinatorial designs are introduced.In Chapter 2,some known results of locally repairable codes are listed.In Chapter 3,the combinatorial characterization of the optimal[k+(?),k,δ]q locally repairable codes with information(r,δ;1)c-locality is given,and five configurations that are commonly used for constructing packings are proposed,then the optimal[k+(?),k,3]q locally repairable codes with information(r,3;1)c-locality are obtained by using the existence of packings when r≥4. |
