Multilevel diversity coding with regeneration

The digital contents in large distributed storage systems may have different reliability and access delay requirements, and for this reason, erasure codes with different strengths need to be utilized to achieve the best storage efficiency. At the same time, in such large distributed storage systems, nodes fail on a regular basis, and the contents stored on them need to be regenerated and stored on other healthy nodes. We formulate the problem of multilevel diversity coding with regeneration to address these considerations, for which the storage vs. repair-bandwidth tradeoff is investigated. We show that the extreme point on this tradeoff corresponding to the minimum possible storage can be achieved by a simple coding scheme, where contents with different reliability requirements are encoded separately using individual regenerating codes without any mixing. On the other hand, we completely characterize the optimal storage-repairbandwidth tradeoff for the case of four storage nodes, and show that a non-vanishing gap exists between the optimal tradeoffs of mixing and non-mixing solutions.