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, $\partial - 1$ erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to ${\rm d}^c_j - 1$ erasures in every window of $j+1$ consecutive blocks of $n$ nodes, where ${\rm d}^c_j$ 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 $n(j+1) - {\rm d}^c_j + 1$ nodes, plus less than $\mu n$ other nodes, in the storage system, where $\mu$ is the memory of the code. A Singleton-type bound is provided for ${\rm d}^c_j$. 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 $n(j+1)$. 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.