Repair Optimal Erasure Codes Through Hadamard Designs

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.