We study the exact and optimal repair of multiple failures in codes for distributed storage. More particularly, we examine the use of interference alignment to build exact scalar minimum storage coordinated regenerating codes (MSCR). We show that it is possible to build codes for the case of k = 2 and d > k by aligning interferences independently but that this technique cannot be applied as soon as k > 2 and d > k. Our results also apply to adaptive regenerating codes.
When there are multiple storage node failures in distributed storage system, regenerating them individually is suboptimal as far as repair bandwidth minimization is concerned. The tradeoff between storage and repair bandwidth is derived in the case where data exchange among the newcomers is enabled. The tradeoff curve with cooperation is strictly better than the one without cooperation. An explicit construction of cooperative regenerating code is given.  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. 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].