Locally Repairable Convolutional Codes With Sliding Window Repair

Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure (or more generally, ∂ - 1 erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to d
_j
^c
-1 erasures in every window of j + 1 consecutive blocks of n nodes, where d
_j
^c
is the jth 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
_j
^c
+1 nodes, plus less than μn other nodes, in the storage system, where μ is the memory of the code. A Singleton-type bound is provided for dcj. 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). In addition, the LRCC is able to perform the flexible and somehow local sliding-window repair by adjusting j. Furthermore, by adjusting and/or sliding the window, the LRCC can potentially correct more erasures in the original window of n(j + 1) nodes than an optimal locally repairable block code of the same rate and locality, and length n(j+1). 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.