Nested erasure codes to achieve the singleton bounds

This paper presents innovative ideas to construct maximum distance separable (MDS) codes, optimal error correction codes that achieve the singleton bounds. The proposed codes are based on a circularly symmetric construction applied to a novel class of nested graphs, referred to as complete-graph-of-rings (CGR). We demonstrate the general idea of transforming graphs to array codes, provide the sufficient conditions for achieving MDS, and present the specific algorithms to construct CGR graphs and MDS-CGR codes. The new codes require minimal encoding and decoding complexity that is theoretically possible, and are particularly useful for disk array applications.