An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes.

Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. In particular, an $(n, k)$ maximum distance separable (MDS) code stores k$ symbols in n$ disks such that the overall system is tolerant to a failure of up to n-k$ disks. However, access to at least k$ disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length \ell$ . The MDS array codes have the potential to repair a single erasure using a fraction 1/(n-k)$ of data stored in the remaining disks. We introduce new methods of analysis, which capitalize on the translation of the storage system problem into a geometric problem on a set of operators and subspaces. In particular, we ask the following question: for a given (n, k)$ , what is the minimum vector-length or subpacketization factor \ell$ required to achieve this optimal fraction? For exact recovery of systematic disks in an MDS code of low redundancy, i.e., k/n>1/2$ , the best known explicit codes have a subpacketization factor \ell$ , which is exponential in k , for an arbitrary number of parity nodes $r=n-k$ , where $\delta=r/(r-1)$ . [ABSTRACT FROM AUTHOR]