Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.
From the opposite point of view, nontrivial upper bounds for exact regenerating codes are studied for example in @cite_22 and @cite_8 . In the case @math , the capacity curve for all possible values of @math was solved in @cite_6 . In the nonasymptotic case, it was shown in @cite_20 that in most points between MBR and MSR, there do not exist exact regenerating codes that would achieve the capacity of functionally repairing regenerating codes.
@cite_20 @cite_22 @cite_6 @cite_8
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In this paper, distributed storage systems with exact repair are studied. Constructions for exact-regenerating codes between the minimum storage regenerating (MSR) and the minimum bandwidth regenerating (MBR) points are given. To the best of our knowledge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in the works by and In contrast to their works, the methods used here are elementary. In this paper, it is shown that in the case that the parameters (n ) , (k ) , and (d ) are close to each other, the given construction is close to optimal when comparing with the known functional repair capacity. This is done by showing that when the distances of the parameters (n ) , (k ) , and (d ) are fixed but the actual values approach to infinity, the fraction of the performance of constructed codes with exact repair and the known capacity of codes with functional repair, approaches to one. Also, a simple variation of the constructed codes with almost the same performance is given. Also some bounds for the capacity of exact-repairing codes are given. These bounds illustrate the relationships between storage codes with different parameters. We give an explicit construction of exact cooperative regenerating codes at the MBCR (minimum bandwidth cooperative regeneration) point. Before the paper, the only known explicit MBCR codes are given with parameters n = d + r and d = k, while our construction applies to all possible values of n, k, d, r. The code has a brief expression in the polynomial form and the data reconstruction is accomplished by bivariate polynomial interpolation. It is a scalar code and operates over a finite field of size q ≥ n. Besides, we establish several subspace properties for linear exact MBCR codes. Based on these properties we prove that linear exact MBCR codes cannot achieve repair-by-transfer. In this paper, we revisit the problem of characterizing the secrecy capacity of minimum storage regenerating (MSR) codes under the passive @math -eavesdropper model, where the eavesdropper has access to data stored on @math nodes and the repair data for an additional @math nodes. We study it from the information-theoretic perspective. First, some general properties of MSR codes as well as a simple and generally applicable upper bound on secrecy capacity are given. Second, a new concept of stable MSR codes is introduced, where the stable property is shown to be closely linked with secrecy capacity. Finally, a comprehensive and explicit result on secrecy capacity in the linear MSR scenario is present, which generalizes all related works in the literature and also predicts certain results for some unexplored linear MSR codes. 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.