Optimal Exact Repair Strategy for the Parity Nodes of the (k+2,k)(k+2,k) Zigzag Code

In this paper, we reinterprets the $(k+2,k)$ Zigzag code in coding matrix and then propose an optimal exact repair strategy for its parity nodes, whose repair disk I/O approaches a lower bound derived in this paper.Comment: 11 page

A Systematic Construction of MDS Codes with Small Sub-packetization Level and Near Optimal Repair Bandwidth

In the literature, all the known high-rate MDS codes with the optimal repair bandwidth possess a significantly large sub-packetization level, which may prevent the codes to be implemented in practical systems. To build MDS codes with small sub-packetization level, existing constructions and theoretical bounds imply that one may sacrifice the optimality of the repair bandwidth. Partly motivated by the work of Tamo et al. (IEEE Trans. Inform. Theory, 59(3), 1597-1616, 2013), in this paper, we present a powerful transformation that can greatly reduce the sub-packetization level of any MDS codes with respect to the same code length n. As applications of the transformation, four high-rate MDS codes having both small sub-packetization level and near optimal repair bandwidth can be obtained, where two of them are also explicit and the required field sizes are comparable to the code length n. Additionally, we propose another explicit MDS code that have small sub-packetization level, near optimal repair bandwidth, and the optimal update property. The required field size is also comparable to the code length n.Comment: 17 page

A Generic Transformation to Enable Optimal Repair in MDS Codes for Distributed Storage Systems

We propose a generic transformation that can convert any nonbinary $(n=k+r,k)$ maximum distance separable (MDS) code into another $(n,k)$ MDS code over the same field such that 1) some arbitrarily chosen $r$ nodes have the optimal repair bandwidth and the optimal rebuilding access, 2) for the remaining $k$ nodes, the normalized repair bandwidth and the normalized rebuilding access (over the file size) are preserved, 3) the sub-packetization level is increased only by a factor of $r$. Two immediate applications of this generic transformation are then presented. The first application is that we can transform any nonbinary MDS code with the optimal repair bandwidth or the optimal rebuilding access for the systematic nodes only, into a new MDS code which possesses the corresponding repair optimality for all nodes. The second application is that by applying the transformation multiple times, any nonbinary $(n,k)$ scalar MDS code can be converted into an $(n,k)$ MDS code with the optimal repair bandwidth and the optimal rebuilding access for all nodes, or only a subset of nodes, whose sub-packetization level is also optimal.Comment: This paper has been published in IEEE Transactions on Information Theor

A Note on the Transformation to Enable Optimal Repair in MDS Codes for Distributed Storage Systems

For high-rate maximum distance separable (MDS) codes, most early constructions can only optimally repair all the systematic nodes but not for all the parity nodes initially. Fortunately, this issue was firstly solved by Li et al. in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a very powerful transformation that can convert any nonbinary MDS code into another MDS code with desired properties was proposed. However, the transformation does not work for binary MDS codes. In this note, we address this issue by proposing another generic transformation that can convert any (n, k) binary MDS code into a new binary MDS code, which endows any r=n-k chosen nodes with the optimal repair bandwidth and the optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth and the normalized rebuilding access for the remaining k nodes under some conditions. As two immediate algorithms of this transformation, we show that 1) by applying the transformation multiple times, any (n,k) binary MDS code can be converted into an (n,k) binary MDS code with the optimal repair bandwidth and the optimal rebuilding access for all nodes, 2) any binary MDS code with the optimal repair bandwidth or the optimal rebuilding access for the systematic nodes only can be converted into an MDS code with the corresponding repair optimality for all nodes.Comment: 17 page

Cascade Codes For Distributed Storage Systems

A novel coding scheme for exact repair-regenerating codes is presented in this paper. The codes proposed in this paper 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 $(n,k,d)$, 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 $\Theta(n)$, and the (unnormalized) 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 \emph{helper-independent}, that is the data sent from each helper only depends on the identity of the helper and failed nodes, but independent from the identity of other helper nodes participating in the repair process