A code symbol of a linear code is said to have locality r if this symbol could be recovered by at most r other code symbols. An (n,k,r) locally repairable code (LRC) with all symbol locality is a linear code with length n, dimension k, and locality r for all symbols. Recently, there are lots of studies on the bounds and constructions of LRCs, most of which are essentially based on the generator matrix of the linear code. Up to now, the most important bounds of minimum distance for LRCs might be the well-known Singleton-like bound and the Cadambe-Mazumdar bound concerning the field size. In this paper, we study the bounds and constructions of LRCs from views of parity-check matrices. Firstly, we set up a new characterization of the parity-check matrix for an LRC. Then, the proposed parity-check matrix is employed to analyze the minimum distance. We give an alternative simple proof of the well-known Singleton-like bound for LRCs with all symbol locality, and then easily generalize it to a more general bound, which essentially coincides with the Cadambe-Mazumdar bound and includes the Singleton-like bound as a specific case. Based on the proposed characterization of parity-check matrices, necessary conditions of meeting the Singleton-like bound are obtained, which naturally lead to a construction framework of good LRCs. Finally, two classes of optimal LRCs based on linearized polynomial theories and Vandermonde matrices are obtained under the construction framework.
These codes @cite_2 are a class of MDS codes with subpacketization. When a single node failure occures, the repair process involves more than @math nodes and each node transfers a linear combination of the packets it stores, which reduces bandwidths compared to classical MDS codes. See @cite_22 @cite_17 for the construction of regenerating codes and @cite_29 for a survey.
@cite_29 @cite_17 @cite_22 @cite_2
2058863419 2126295689 2150777202 2105185344
Distributed storage systems often introduce redundancy to increase reliability. When coding is used, the repair problem arises: if a node storing encoded information fails, in order to maintain the same level of reliability we need to create encoded information at a new node. This amounts to a partial recovery of the code, whereas conventional erasure coding focuses on the complete recovery of the information from a subset of encoded packets. The consideration of the repair network traffic gives rise to new design challenges. Recently, network coding techniques have been instrumental in addressing these challenges, establishing that maintenance bandwidth can be reduced by orders of magnitude compared to standard erasure codes. This paper provides an overview of the research results on this topic. The high repair cost of (n, k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of MDS codes, called Repair MDS codes, that can significantly reduce repair bandwidth over conventional MDS codes. In this paper, we describe (n, k, d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to d survivor nodes, where k ≤ d ≤ n-1. We construct Exact-Repair MDS codes that are optimal in repair bandwidth for the cases of: (α) k n ≤ 1 2 and d ≥ 2k - 11; (b) k ≤ 3. Our codes are deterministic and require a finite-field size of at most 2(n - k). Our constructive codes are based on interference alignment techniques. Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An [n, k, d] regenerating code permits the data to be recovered by connecting to any k of the n nodes in the network, while requiring that a failed node be repaired by connecting to any d nodes. The amount of data downloaded for repair is typically much smaller than the size of the source data. Previous constructions of exact-regenerating codes have been confined to the case n=d+1 . In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of [n, k, d] and (b) Minimum Storage Regenerating (MSR) codes for all [n, k, d ≥ 2k-2], using a new product-matrix framework. The product-matrix framework is also shown to significantly simplify system operation. To the best of our knowledge, these are the first constructions of exact-regenerating codes that allow the number n of nodes in the network, to be chosen independent of the other parameters. The paper also contains a simpler description, in the product-matrix framework, of a previously constructed MSR code with [n=d+1, k, d ≥ 2k-1]. Distributed storage systems provide reliable access to data through redundancy spread over individually unreliable nodes. Application scenarios include data centers, peer-to-peer storage systems, and storage in wireless networks. Storing data using an erasure code, in fragments spread across nodes, requires less redundancy than simple replication for the same level of reliability. However, since fragments must be periodically replaced as nodes fail, a key question is how to generate encoded fragments in a distributed way while transferring as little data as possible across the network. For an erasure coded system, a common practice to repair from a single node failure is for a new node to reconstruct the whole encoded data object to generate just one encoded block. We show that this procedure is sub-optimal. We introduce the notion of regenerating codes, which allow a new node to communicate functions of the stored data from the surviving nodes. We show that regenerating codes can significantly reduce the repair bandwidth. Further, we show that there is a fundamental tradeoff between storage and repair bandwidth which we theoretically characterize using flow arguments on an appropriately constructed graph. By invoking constructive results in network coding, we introduce regenerating codes that can achieve any point in this optimal tradeoff.