| In the era of big data,the storage of massive data is inseparable from distributed storage systems.The distributed storage system stores data in a number of nodes,thus get the required data only needs to access some nodes.Locally repairable codes(LRC)have good algebraic structure and the function of repairing data.Locally repairable codes with locality of r can use at most other r bits to repair of single node failure in distributed storage systems.Locally repairable codes with locality of(r,δ)can solve the problem of repairing multiple node failures in the distributed storage systems.Therefore,the locally repairable codes become the main encoding method in distributed storage systems.This paper studies the constructions of locally repairable codes over the finite field Fq.The specific contents are as follows:On the one hand,we construct the optimal locally repairable codes with locality of r over the finite field Fq.First,the optimal locally repairable codes over the finite field Fq with a minimum distance of d=5 and d=6 are constructed by using the parity-check matrix,and then the maximum length of optimal locally repairable codes over the finite field F5 are studied,maximum length of these codes are obtained.On the other hand,we construct the optimal locally repairable codes with locality of(r,δ)over over the finite field Fq.A class of optimal locally repairable codes with minimum distances d=δ+m+1,and two classes of optimal locally repairable codes with lengths which is multiple of q+1,minimum distances d=δ+4 are constructed by using cyclic codes over the finite field Fq. |
