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

Multi-Layer Transformed MDS Codes with Optimal Repair Access and Low Sub-Packetization

An $(n,k)$ maximum distance separable (MDS) code has optimal repair access if the minimum number of symbols accessed from $d$ surviving nodes is achieved, where $k+1\le d\le n-1$. Existing results show that the sub-packetization $\alpha$ of an $(n,k,d)$ high code rate (i.e., $k/n>0.5$) MDS code with optimal repair access is at least $(d-k+1)^{\lceil\frac{n}{d-k+1}\rceil}$. In this paper, we propose a class of multi-layer transformed MDS codes such that the sub-packetization is $(d-k+1)^{\lceil\frac{n}{(d-k+1)\eta}\rceil}$, where $\eta=\lfloor\frac{n-k-1}{d-k}\rfloor$, and the repair access is optimal for any single node. We show that the sub-packetization of the proposed multi-layer transformed MDS codes is strictly less than the existing known lower bound when $\eta=\lfloor\frac{n-k-1}{d-k}\rfloor>1$, achieving by restricting the choice of $d$ specific helper nodes in repairing a failed node. We further propose multi-layer transformed EVENODD codes that have optimal repair access for any single node and lower sub-packetization than the existing binary MDS array codes with optimal repair access for any single node. With our multi-layer transformation, we can design new MDS codes that have the properties of low computational complexity, optimal repair access for any single node, and relatively small sub-packetization, all of which are critical for maintaining the reliability of distributed storage systems