Locally Repairable Convolutional Codes with Sliding Window Repair
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Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure, and sliding-window global repair, which can correct up to d^c_j - 1 node erasures in a window of j+1 consecutive blocks of n nodes, where d^c_j is the j th column distance of the code. The parameter j can be adjusted, for a fixed LRCC, according to different catastrophic erasure patterns, requiring only to contact n(j+1) - d^c_j + 1 nodes, plus less than μ n other nodes, in the storage system, where μ is the memory of the code. A Singleton-type bound is provided for d^c_j . If it attains such a bound, an LRCC can correct the same number of catastrophic erasures in a window of length n(j+1) as an optimal locally repairable block code of the same rate and locality, and with block length n(j+1) , but being able to perform the flexible and somehow local sliding-window repair by adjusting j . Furthermore, by sliding the window to consider previous or consequent nodes without erasures, or by increasing the window size, the LRCC can potentially correct more erasures in the original window of n(j+1) nodes than the optimal locally repairable block code. Finally, the concept of partial maximum distance profile (partial MDP) codes is introduced. Partial MDP codes can correct all information-theoretically correctable erasure patterns for a given locality, local distance and information rate. An explicit construction of partial MDP codes whose column distances attain the provided Singleton-type bound, up to certain parameter j=L , is obtained based on known maximum sum-rank distance convolutional codes.
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