Abstract-Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An regenerating code permits the data to be recovered by connecting to any of the nodes in the network, while requiring that a failed node be repaired by connecting to any 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 . In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of and (b) Minimum Storage Regenerating (MSR) codes for all , 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 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 .

Abstract-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, Wu et al. 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.

Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any subset of k nodes within the n-node network. However, regenerating codes possess in addition, the ability to repair a failed node by connecting to an arbitrary subset of d nodes. It has been shown that for the case of functional-repair, there is a tradeoff between the amount of data stored per node and the bandwidth required to repair a failed node. A special case of functional-repair is exact-repair where the replacement node is required to store data identical to that in the failed node. Exact-repair is of interest as it greatly simplifies system implementation.The first result of the paper is an explicit, exact-repair code for the point on the storage-bandwidth tradeoff corresponding to the minimum possible repair bandwidth, for the case when d = n − 1. This code has a particularly simple graphical description and most interestingly, has the ability to carry out exact-repair through mere transfer of data and without any need to perform arithmetic operations. Hence the term repair-by-transfer.The second result of this paper shows that the interior points on the storage-bandwidth tradeoff cannot be achieved under exact-repair, thus pointing to the existence of a separate tradeoff under exact-repair. Specifically, we identify a set of scenarios, termed 'helper node pooling', and show that it is the necessity to satisfy such scenarios that over-constrains the system.

Private information retrieval (PIR) systems allow a user to retrieve a record from a public database without revealing to the server which record is being retrieved. The literature on PIR considers only replication-based systems, wherein each storage node stores a copy of the entire data. However, systems based on erasure codes are gaining increasing popularity due to a variety of reasons. This paper initiates an investigation into PIR in erasure-coded systems by establishing its capacity and designing explicit codes and algorithms. The notion of privacy considered here is information-theoretic, and the metric optimized is the amount of data downloaded by the user during PIR.In this paper, we present four main results. First, we design an explicit erasure code and PIR algorithm that requires only one extra bit of download to provide perfect privacy. In contrast, all existing PIR algorithms require a download of at least twice the size of the requisite data. Second, we derive lower bounds proving the necessity of downloading at least one additional bit. This establishes the precise capacity of PIR with respect to the metric of download. These results are also applicable to PIR in replication-based systems, which are a special case of erasure codes. Our third contribution is a negative result showing that capacity-achieving codes necessitate super-linear storage overheads. This motivates the fourth contribution of this paper: an erasure code and PIR algorithm that requires a linear storage overhead, provides high reliability to the data, and is a small factor away from the capacity.

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 exactrepair 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 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 non-existence 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.