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Linear programming bounds for distributed storage codes

This work was partially supported by a grant from Australian Research Council (DP150103658) and the Research Grants Council of the Hong Kong Special Administrative Region, China under Project CityU 11205318

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  • A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to have multiple repair groups. In this paper we consider robust locally repairable coding schemes which guarantee that there exist multiple distinct (not necessarily disjoint) alternative local repair groups for any single failure such that the failed node can still be repaired locally even if some of the repair groups are not available. We use linear programming techniques to establish upper bounds on the size of these codes. We also provide two examples of robust locally repairable codes that are optimal regarding our linear programming bound. Furthermore, we address the update efficiency problem of the distributed data storage networks. Any modification on the stored data will result in updating the content of the storage nodes. Therefore, it is essential to minimise the number of nodes which need to be updated by any change in the stored data. We characterise the update-efficient storage code properties and establish the necessary conditions of existence update-efficient locally repairable storage codes.

    Mathematics Subject Classification: Primary: 68P20, 94B05, 68P30; Secondary: 90C05.

    Citation:

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  • Figure 1.  Generator matrix of an update efficient storage code

    Figure 2.  Parity check matrix of an update efficient storage code

    Figure 3.  A (3, 3, 1, 1) robust locally repairable code of length $ {N} = 16 $ and dimension $ {K} = 9 $

    Figure 4.  Upper bounds for $ (r,3,\Gamma,\zeta) $ binary robust locally repairable code of length $ N = 16 $

    Figure 5.  A robust locally repairable code of length $ N = 8 $ and dimension $ K = 4 $ with $ \zeta = 7 $ repair groups for any failure

    Figure 6.  Upper bounds for $ (r,3,\Gamma,\zeta) $ binary robust locally repairable code of length $ N = 8 $

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