Repair Optimal Erasure Codes through Hadamard Designs

TL;DR

This paper introduces a novel explicit construction of high-rate MDS erasure codes using Hadamard matrices, achieving optimal repair bandwidth for all single node failures in distributed storage systems.

Contribution

It presents the first explicit 2-parity MDS code with optimal repair properties for all single node failures, utilizing Hadamard designs and perfect interference alignment.

Findings

Constructed the first explicit 2-parity MDS code with optimal repair bandwidth.

Generalized the construction to m-parity MDS codes with optimal repair.

Established connections between the new codes and existing permutation-matrix based codes.

Abstract

In distributed storage systems that employ erasure coding, the issue of minimizing the total {\it 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. In this work, we use Hadamard matrices to construct the first explicit 2-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 file size, or number of extensions. We generalize this construction to design $m$-parity MDS codes that achieve the optimum repair communication for single systematic node failures and show that there is an interesting connection between our $m$-parity codes and the systematic-repair optimal permutation-matrix based codes of Tamo {\it et al.} \cite{Tamo} and Cadambe {\it et al.} \cite{PermCodes_ISIT, PermCodes}.