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 dcj−1 erasures in every window of j+1 consecutive blocks of n nodes, where dcj−1 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)−dcj+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−1 . 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.This work was supported in part by the Independent Research Fund Denmark under Grant DFF-7027-00053B, in part by the Generalitat Valenciana under Grant AICO/2017/128, and in part by the Universitat d’Alacant under Grant VIGROB-287