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Repair Optimal Erasure Codes Through Hadamard Designs

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3 Author(s)
Papailiopoulos, D.S. ; Electrical Engineering, University of Southern California, ; Dimakis, A.G. ; Cadambe, V.R.

In distributed storage systems that employ erasure coding, the issue of minimizing the total communication required to exactly rebuild a storage node after a failure arises. This repair bandwidth depends on the structure of the storage code and the repair strategies used to restore the lost data. Designing high-rate maximum-distance separable (MDS) codes that achieve the optimum repair communication has been a well-known open problem. Our work resolves, in part, this open problem. In this study, we use Hadamard matrices to construct the first explicit two-parity MDS storage code with optimal repair properties for all single node failures, including the parities. Our construction relies on a novel method of achieving perfect interference alignment over finite fields with a finite number of symbol extensions. We generalize this construction to design m -parity MDS codes that achieve the optimum repair communication for single systematic node failures.

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Information Theory, IEEE Transactions on  (Volume:59 ,  Issue: 5 )