Regenerating code is a class of code very suitable for distributed storage systems, which can maintain optimal bandwidth and storage space. Two types of important regenerating code have been constructed: the minimum storage regeneration (MSR) code and the minimum bandwidth regeneration (MBR) code. However, in hostile networks where adversaries can compromise storage nodes, the storage capacity of the network can be significantly affected. In this paper, we propose two optimal constructions of regenerating codes through rate-matching that can combat against this kind of adversaries in hostile networks: 2-layer rate-matched regenerating code and @math -layer rate-matched regenerating code. For the 2-layer code, we can achieve the optimal storage efficiency for given system requirements. Our comprehensive analysis shows that our code can detect and correct malicious nodes with higher storage efficiency compared to the universally resilient regenerating code which is a straightforward extension of regenerating code with error detection and correction capability. Then we propose the @math -layer code by extending the 2-layer code and achieve the optimal error correction efficiency by matching the code rate of each layer's regenerating code. We also demonstrate that the optimized parameter can achieve the maximum storage capacity under the same constraint. Compared to the universally resilient regenerating code, our code can achieve much higher error correction efficiency.
The regenerating code can be divided into functional regeneration and exact regeneration. In the functional regeneration, the replacement node regenerates a new component that can functionally replace the failed component instead of being the same as the original stored component. @cite_24 formulated the data regeneration as a multicast network coding problem and constructed functional regenerating codes. @cite_25 implemented a random linear regenerating codes for distributed storage systems. @cite_18 proved that by allowing data exchange among the replacement nodes, a better tradeoff between repair bandwidth @math and per node storage @math can be achieved. In the exact regeneration, the replacement node regenerates the exact symbols of a failed node. @cite_32 proposed to reduce the regeneration bandwidth through algebraic alignment. @cite_4 provided a code structure for exact regeneration using interference alignment technique. @cite_14 presented optimal exact constructions of MBR codes and MSR codes under product-matrix framework. This is the first work that allows independent selection of the nodes number @math in the network.
@cite_18 @cite_14 @cite_4 @cite_32 @cite_24 @cite_25
2111915261 2150777202 2056826630 2108897603
Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any arbitrary k of n nodes. However regenerating codes possess in addition, the ability to repair a failed node by connecting to any arbitrary d nodes and downloading an amount of data that is typically far less than the size of the data file. This amount of download is termed the repair bandwidth. Minimum storage regenerating (MSR) codes are a subclass of regenerating codes that require the least amount of network storage; every such code is a maximum distance separable (MDS) code. Further, when a replacement node stores data identical to that in the failed node, the repair is termed as exact. The four principal results of the paper are (a) the explicit construction of a class of MDS codes for d = n - 1  2k - 1 termed the MISER code, that achieves the cut-set bound on the repair bandwidth for the exact repair of systematic nodes, (b) proof of the necessity of interference alignment in exact-repair MSR codes, (c) a proof showing the impossibility of constructing linear, exact-repair MSR codes for d <; 2k - 3 in the absence of symbol extension, and (d) the construction, also explicit, of high-rate MSR codes for d = k + 1. Interference alignment (IA) is a theme that runs throughout the paper: the MISER code is built on the principles of IA and IA is also a crucial component to the nonexistence proof for d <; 2k - 3. To the best of our knowledge, the constructions presented in this paper are the first explicit constructions of regenerating codes that achieve the cut-set bound. 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]. We consider the setting of data storage across n nodes in a distributed manner. A data collector (DC) should be able to reconstruct the entire data by connecting to any k out of the n nodes and downloading all the data stored in them. When a node fails, it has to be regenerated back using the existing nodes. In a recent paper, have obtained an information theoretic lower bound for the repair bandwidth. Also, there has been additional interest in storing data in systematic form as no post processing is required when DC connects to k systematic nodes. Because of their preferred status there is a need to regenerate back any systematic node quickly and exactly. Replacement of a failed node by an exact replica is termed Exact Regeneration.In this paper, we consider the problem of minimizing the repair bandwidth for exact regeneration of the systematic nodes. The file to be stored is of size B and each node can store alpha = B k units of data. A failed systematic node is regenerated by downloading beta units of data each from d existing nodes. We give a lower bound for the repair bandwidth for exact regeneration of the systematic nodes which matches with the bound given by For d >= 2k-1 we give an explicit code construction which minimizes the repair bandwidth when the existing k-1 systematic nodes participate in the regeneration. We show the existence and construction of codes that achieve the bound for d >= 2k-3. Here we also establish the necessity of interference alignment. We prove that the bound is not achievable for d = 2k-1. Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A subspace based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular. In this paper, we provide explicit constructions for a class of exact-repair regenerating codes that possess a layered structure. These regenerating codes correspond to interior points on the storage-repair-bandwidth tradeoff where the cut-set bound of network coding is known to be not achievable under exact repair. The codes presented in this paper compare very well in comparison to schemes that employ space-sharing between MSR and MBR points, and come closest of all-known explicit constructions to interior points of the tradeoff. The codes can be constructed for a wide range of parameters, are high-rate, can repair multiple nodes simultaneously and no computation at helper nodes is required to repair a failed node. We also construct optimal codes with locality in which the local codes are layered regenerating codes. In distributed storage systems, regenerating codes can achieve the optimal tradeoff between storage capacity and repair bandwidth. However, a critical drawback of existing regenerating codes, in general, is the high coding and repair complexity, since the coding and repair processes involve expensive multiplication operations in finite field. In this paper, we present a design framework of regenerating codes, which employ binary addition and bitwise cyclic shift as the elemental operations, named BASIC regenerating codes. The proposed BASIC regenerating codes can be regarded as a concatenated code with the outer code being a binary parity-check code, and the inner code being a regenerating code utilizing the binary parity-check code as the alphabet. We show that the proposed functional-repair BASIC regenerating codes can achieve the fundamental tradeoff curve between the storage and repair bandwidth asymptotically of functional-repair regenerating codes with less computational complexity. Furthermore, we demonstrate that the existing exact-repair product-matrix construction of regenerating codes can be modified to exact-repair BASIC product-matrix regenerating codes with much less encoding, repair, and decoding complexity from the theoretical analysis, and with less encoding time, repair time, and decoding time from the implementation results.