Pyramid codes: Flexible schemes to trade space for access efficiency in reliable data storage systems

To flexibly explore the trade-offs between storage space and access efficiency in reliable data storage Systems, we describe two classes of erasure resilient coding schemes: basic and generalized Pyramid Codes. The basic Pyramid Codes can be simply derived from any existing codes, and thus all known efficient encoding/decoding techniques directly apply. The generalized Pyramid Codes are radically advanced new codes, which can further improve access efficiency and/or reliability upon the basic Pyramid Codes. We also establish a necessary condition for any failure pattern to be ever recoverable, and show that the generalized Pyramid Codes are optimal in failure recovery (i.e., the necessary condition is also sufficient, and any failure pattern that is ever recoverable can indeed be recovered).