A New Class of MDS Erasure Codes Based on Graphs

TL;DR

This paper introduces a novel class of MDS erasure codes derived from nested graphs called complete-graph-of-rings, offering a systematic construction method with minimal encoding complexity for use in disk arrays.

Contribution

It develops a new graph-based method to construct MDS array codes, linking graph theory with coding, and introduces CGR codes that generalize B-codes with simple encoding and decoding.

Findings

CGR codes are MDS and optimal for disk arrays.

CGR codes subsume B-codes as contracted forms.

CGR codes require minimal encoding and decoding complexity.

Abstract

Maximum distance separable (MDS) array codes are XOR-based optimal erasure codes that are particularly suitable for use in disk arrays. This paper develops an innovative method to build MDS array codes from an elegant class of nested graphs, termed \textit{complete-graph-of-rings (CGR)}. We discuss a systematic and concrete way to transfer these graphs to array codes, unveil an interesting relation between the proposed map and the renowned perfect 1-factorization, and show that the proposed CGR codes subsume B-codes as their "contracted" codes. These new codes, termed \textit{CGR codes}, and their dual codes are simple to describe, and require minimal encoding and decoding complexity.