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Regenerating codes for distributed storage have attracted much researchinterest in the past decade. Such codes trade the bandwidth needed to repair afailed node with the overall amount of data stored in the network. Minimumstorage regenerating (MSR) codes are an important class of optimal regeneratingcodes that minimize (first) the amount of data stored per node and (then) therepair bandwidth. Specifically, an $[n,k,d]$-$(\alpha)$ MSR code $\mathbb{C}$over $\mathbb{F}_q$ is defined as follows. Using such a code $\mathbb{C}$, afile $\cal{F}$ consisting of $\alpha k$ symbols over $\mathbb{F}_q$ can bedistributed among $n$ nodes, each storing $\alpha$ symbols, in such a way that:The file $\cal{F}$ can be recovered by downloading the content of any $k$ ofthe $n$ nodes; and the content of any failed node can be reconstructed byaccessing any $d$ of the remaining $n-1$ nodes and downloading $\alpha/(d-k+1)$symbols from each of these nodes. Unfortunately, explicit constructions of$[n,k,d]$ MSR codes are known only for certain special cases: either low rate,namely $k/n<0.5$, or high repair connectivity, namely $d = n-1$. Althoughsetting $d = n-1$ minimizes the repair bandwidth, it may be impractical toconnect to all the remaining nodes in order to repair a single failed node. Ourmain result in this paper is an explicit construction of systematic-repair$[n,k,d]$ MSR codes for all possible values of parameters $n,k,d$. Inparticular, we construct systematic-repair MSR codes of high rate $k/n>0.5$ andlow repair connectivity $k< d<n-1$. Such codes were not previously known toexist. In order to construct these codes, we solve simultaneously severalrepair scenarios, each of which is expressible as an interference alignmentproblem. Extension of our results beyond systematic repair remains an openproblem.

Minimum Storage Regenerating Codes For All Parameters