Belief Propagation Decodable XOR based Erasure Codes For Distributed Storage Systems

LDPC codes and digital fountain techniques have received significant attention from both academics and industry in the past few years. There have also been extensive interests in applying LDPC code techniques to distributed storage systems such as cloud data storage in recent years. This paper carries out the theoretical analysis on the feasibility and performance issues for applying LT codes to distributed storage systems. By employing the underlying ideas of efficient Belief Propagation (BP) decoding process in LT codes, this paper introduces two classes of codes called flat BP-XOR codes and array BP-XOR codes. We will show the equivalence between the edge-colored graph model and degree-one-and-two encoding symbols based array BP-XOR codes. Using this equivalence result, we are able to design novel [n, n− 2, 3] and [n, 2, n− 2 + 1] array BP-XOR codes. In this paper, we will also use companion matrices of low weight irreducible binary polynomials to design general MDS [n, k] array codes. Our results shows that the Vandermonde parity check matrices have 10% to 30% less density than Cauchy parity check matrices. Since the density of parity check matrices of array codes directly translates to the update complexity and degraded reads performance in cloud storage systems, our erasure code design is better suited to distributed cloud storage systems.