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Maximum distance separable (MDS) codes are optimal error-correcting codes inthe sense that they provide the maximum failure-tolerance for a given number ofparity nodes. Suppose that an MDS code with $k$ information nodes and $r=n-k$parity nodes is used to encode data in a distributed storage system. It isknown that if $h$ out of the $n$ nodes are inaccessible and $d$ surviving(helper) nodes are used to recover the lost data, then we need to download atleast $h/(d+h-k)$ fraction of the data stored in each of the helper nodes(Dimakis et. al., 2010 and Cadambe et al., 2013). If this lower bound isachieved for the repair of any $h$ erased nodes from any $d$ helper nodes, wesay that the MDS code has the $(h,d)$-optimal repair property.  We study high-rate MDS array codes with the optimal repair property. Explicitconstructions of such codes in the literature are only available for the caseswhere there are at most 3 parity nodes, and these existing constructions canonly optimally repair a single node failure by accessing all the survivingnodes.  In this paper, given any $r$ and $n$, we present two explicit constructionsof MDS array codes with the $(h,d)$-optimal repair property for all $h\le r$and $k\le d\le n-h$ simultaneously. Codes in the first family can beconstructed over any base field $F$ as long as $|F|\ge sn,$ where$s=\text{lcm}(1,2,\dots,r).$ The encoding, decoding, repair of failed nodes,and update procedures of these codes all have low complexity. Codes in thesecond family have the optimal access property and can be constructed over anybase field $F$ as long as $|F|\ge n+1.$ Moreover, both code families have theoptimal error resilience capability when repairing failed nodes. We alsoconstruct several other related families of MDS codes with the optimal repairproperty.

Explicit constructions of high-rate MDS array codes with optimal repair bandwidth