Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. In particular, an (n, κ) 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 The MDS array codes have the potential to repair a single erasure using a fraction 1/(n ?κ) 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, κ), what is the minimum vector-length or subpacketization factor - required to achieve this optimal fraction? For exact recovery of systematic disks in an MDS code of low redundancy, i.e., κ/n > 1/2, the best known explicit codes have a subpacketization factor , which is exponential in k. It has been conjectured that for a fixed number of parity nodes, it is in fact necessary for to be exponential in k. In this paper, we provide a new log-squared converse bound on k for a given -, and prove that k ≤ 2 log2 (logδ +1), for an arbitrary number of parity nodes r = n ? k, where δ = r/(r ? 1).
- distributed storage
- error correction codes
- interference alignment
All Science Journal Classification (ASJC) codes
- Information Systems
- Computer Science Applications
- Library and Information Sciences