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