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Two widely studied models of multiple-node repair in distributed storagesystems are centralized repair and cooperative repair. The centralized modelassumes that all the failed nodes are recreated in one location, while thecooperative one stipulates that the failed nodes may communicate but aredistinct, and the amount of data exchanged between them is included in therepair bandwidth.  As our first result, we prove a lower bound on the minimum bandwidth ofcooperative repair. We also show that the cooperative model is stronger thanthe centralized one, in the sense that any MDS code with optimal repairbandwidth under the former model also has optimal bandwidth under the latterone. These results were previously known under the additional "uniformdownload" assumption, which is removed in our proofs.  As our main result, we give explicit constructions of MDS codes with optimalcooperative repair for all possible parameters. More precisely, given any$n,k,h,d$ such that $2\le h \le n-d\le n-k$ we construct $(n,k)$ MDS codes overthe field $F$ of size $|F|\ge (d+1-k)n$ that can optimally repair any $h$erasures from any $d$ helper nodes. The repair scheme of our codes involves tworounds of communication. In the first round, each failed node downloadsinformation from the helper nodes, and in the second one, each failed nodedownloads additional information from the other failed nodes. This implies thatour codes achieve the optimal repair bandwidth using the smallest possiblenumber of rounds.

Cooperative repair: Constructions of optimal MDS codes for all admissible parameters