A novel coding scheme for exact repair-regenerating codes is presented in this paper. The codes proposed in this work can trade between the repair bandwidth of nodes (number of downloaded symbols from each surviving node in a repair process) and the required storage overhead of the system. These codes work for general system parameters (n, k, d), which are the total number of nodes, the number of nodes suffice for data recovery, and the number of helper nodes in a repair process, respectively. The proposed construction offers a unified scheme to develop exact-repair regenerating codes for the entire trade-off, including the MBR and MSR points. We conjecture that the new storage-vs.-bandwidth trade-off achieved by the proposed codes is optimum. Some other key features of this code include: the construction is linear; the required field size is only Θ(n); and the code parameters and in particular sub-packetization level is at most (d - k +1)k; which is independent of the number of the parity nodes. Moreover, the proposed repair mechanism is helperindependent, that is the data sent from each helper only depends on the identity of the helper and failed nodes, but independent of the identity of other helper nodes participating in the repair process.

This paper investigates the problem of how to efficiently operate Gaussian half-duplex diamond networks with N relays. It derives sufficient conditions that ensure that the network is operated close to its Shannon capacity, with a linear number in N (instead of exponential) of receive/transmit configuration states. Particularly, these states consist of having either at most one relay receiving or at most one relay transmitting. A transmission scheme is also designed and it is shown that, when the aforementioned conditions are satisfied, it achieves a rate that is to within a constant gap of the Shannon capacity. An appealing feature of the proposed scheme is that it offers guidelines on how to route the information through the relays so that the network operates close to its Shannon capacity.

Determinant codes are a class of exact-repair regenerating codes for distributed storage systems with parameters (n, k = d, d). These codes cover the entire trade-off between per-node storage and repair-bandwidth. In an earlier work of the authors, the repair data of the determinant code sent by a helper node to repair a failed node depends on the identity of the other helper nodes participating in the process, which is practically undesired. In this paper, a new repair mechanism is proposed for determinant codes, which relaxes this dependency, while preserving all other properties of the code. Moreover, it is shown that the determinant codes are capable of repairing multiple failures, with a per-node repair-bandwidth which scales sub-linearly with the number of failures.

A novel class of exact-repair regenerating codes is introduced for a distributed storage system with arbitrary parameters (n, k, d). The proposed construction is based on the optimum determinant codes for (n, k=d, d) systems. This construction yields an achievable trade-off between the storage and the repair bandwidth, consisting of k corner points, which meets the optimum trade-off at the MBR and MSR points, and improves all the previously known bounds for interior points. The sub-packetization level of the proposed code only depends on k and d, but not number of nodes n. Further, the required field size for the proposed code is Θ(n). We conjecture that the proposed codes can universally achieve the optimum trade-off.

This paper focuses on the design of regeneration codes. An (n, k, d) exact-regenerating code encodes and stores the data into n nodes such that the entire data can be recovered from any k nodes, and the missing coded information of any failed node can be identically recovered by the help of d nodes. In an earlier work of the authors, determinant codes are introduced for any (n, k, d = k) system, and they are shown to achieve the optimum tradeoff between the node storage α and the repair-bandwidth β In this work, the latter constraint of d = k is relaxed, and the construction of determinant codes is generalized to arbitrary parameters (n, k, d), for a certain range of (α, β). The proposed construction is scalable, in the sense that the system performance only depend on k and d, and the same of operating point (α, β) can be universally achieved for any number of nodes n. The resulting codes are linear, and the required size of the underlying finite field is not greater than Θ(n).

The exact-repair regeneration codes for distributed storage system are studied in this work. A novel coding scheme is proposed for code construction for any (n, k, d = k) system, and it is shown to be optimal. In particular, the optimum tradeoff of exact-repair regeneration system is fully characterized for any system with d = k. The new construction is based on fundamental properties of matrix determinant, thus the code is called determinant code. It is devised for the entire range of (α, β) on the optimum tradeoff.

The exact-repair regenerating codes for distributed storage system are studied in this work. A novel coding scheme is proposed for code construction for any (n, k, d = k) system. It is shown that the proposed codes are optimum, in the sense that any operating point satisfying the lower bound for the storage-bandwidth trade-off can be achieved with the proposed construction. As a consequence, the optimum linear trade-off of exact-regenerating system is fully characterized for any systems with d = k. The proposed codes are linear, and can be generated by multiplication of an encoder matrix with a so-called data matrix. The exact regenerating property is provided based on fundamental properties of matrix determinants, and in particular, generalized Laplace expansion for determinant. Thus the code is called determinant code. Importantly, the field size required for this code construction is 8(n), the total number of nodes in the system.

Characterizing the exact repair storage-vs-repair bandwidth tradeoff for distributed storage systems remains an open problem for more than four storage nodes. Motivated by the prevalence and practical applicability of linear codes, the exact repair problem when restricted to linear codes is considered. The main result of this paper is a new approach to develop bounds for exact repair distributed storage systems with linear codes (LDSS). Using this approach, the exact repair region for the (k + 1, k, k) LDSS is completely characterized. The new approach utilizes the properties of linear codes together with the exact repair constraints. These constraints are formally captured through an optimization problem with a recursive structure, and its solution finally yields the new bounds for the LDSS. These bounds together with recent code constructions characterize the exact repair region for (k + 1, k, k) LDSS.