text: 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.
  @cite_20: Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any subset of nodes within the -node network. However, regenerating codes possess in addition, the ability to repair a failed node by connecting to an arbitrary subset of 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 this 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 . This code has a particularly simple graphical description, and most interestingly has the ability to carry out exact repair without any need to perform arithmetic operations. We term this ability of the code to perform repair through mere transfer of data as 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 which we term as “helper node pooling,” and show that it is the necessity to satisfy such scenarios that overconstrains the system.
  @cite_22: While the tradeoff between the amount of data stored and the repair bandwidth of an (n, k, d) regenerating code has been characterized under functional repair (FR), the case of exact repair (ER) remains unresolved. It is known that there do not exist ER codes which lie on the FR tradeoff at most of the points. The question as to whether one can asymptotically approach the FR tradeoff was settled recently by Tian who showed that in the (4, 3, 3) case, the ER region is bounded away from the FR region. The FR tradeoff serves as a trivial outer bound on the ER tradeoff. In this paper, we extend Tian's results by establishing an improved outer bound on the ER tradeoff which shows that the ER region is bounded away from the FR region, for any (n; k; d). Our approach is analytical and builds upon the framework introduced earlier by Shah et. al. Interestingly, a recently-constructed, layered regenerating code is shown to achieve a point on this outer bound for the (5, 4, 4) case. This represents the first-known instance of an optimal ER code that does not correspond to a point on the FR tradeoff.
  @cite_6: Exact-repair regenerating codes are considered for the case (n, k, d) = (4, 3,3), for which a complete characterization of the rate region is provided. This characterization answers in the affirmative the open question whether there exists a non-vanishing gap between the optimal bandwidth-storage tradeoff of the functional-repair regenerating codes (i.e., the cut-set bound) and that of the exact-repair regenerating codes. The converse proof relies on the existence of symmetric optimal solutions. For the achievability, only one non-trivial corner point of the rate region needs to be addressed, for which an explicit binary code construction is given.
  @cite_8: We address the open problem of establishing the rate region for exact-repair regenerating codes for given parameters (n,k,d). Tian determined the rate region for a (4,3,3) code and found that it lies strictly within the functional-repair rate region. Using different methods, Sasidharan, Senthoor and Kumar proved a non-vanishing gap between the functional-repair outer bound and the exact-repair outer bound for codes with k>=3. Our main results are two improved outer bounds for exact-repair regenerating codes. They capture and then extend essential parts in the proofs by Tian and by Sasidharan, Senthoor and Kumar. We show that the bounds can be combined for further improvements.
summary: 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.