TY - JOUR

T1 - An improved sub-packetization bound for minimum storage regenerating codes

AU - Goparaju, Sreechakra

AU - Tamo, Itzhak

AU - Calderbank, Robert

PY - 2014/5

Y1 - 2014/5

N2 - 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).

AB - 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).

KW - distributed storage

KW - error correction codes

KW - interference alignment

KW - sub-packetization

UR - http://www.scopus.com/inward/record.url?scp=84899625009&partnerID=8YFLogxK

U2 - 10.1109/TIT.2014.2309000

DO - 10.1109/TIT.2014.2309000

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AN - SCOPUS:84899625009

SN - 0018-9448

VL - 60

SP - 2770

EP - 2779

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 5

M1 - 6750093

ER -