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

Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity 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 is known 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 at least $h/(d+h-k)$ fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and Cadambe et al., 2013). If this lower bound is achieved for the repair of any $h$ erased nodes from any $d$ helper nodes, we say that the MDS code has the $(h,d)$-optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any $r$ and $n$, we present two explicit constructions of 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 be constructed 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 the second family have the optimal access property and can be constructed over any base field $F$ as long as $|F|\ge n+1.$ Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property.