This paper introduces efficient erasure correcting codes based on sparse bipartite graphs. The authors propose a decoding algorithm for codes derived from cascades of sparse bipartite graphs and analyze it using a discrete-time random process. They derive a criterion for successful decoding based on node degree fractions in the graph. The codes can be encoded in time proportional to ln(1/ε) times the block length n, and can recover from a portion of their entries of length (1+ε)Rn with high probability. The decoding algorithm also runs in time proportional to n ln(1/ε). The algorithms have been implemented and work well in practice. The paper discusses the use of erasure channels, large deviation analysis, and low-density parity-check codes. It introduces a new class of codes that are almost MDS (minimum-distance separable) in the sense that they can recover from a random set of k(1+ε) coordinate places with high probability. These codes have linear time encoding and decoding algorithms and can use an arbitrary alphabet size. The authors describe a construction using bipartite graphs, where each check bit is the XOR of its neighboring message bits. They also discuss cascading codes and the use of irregular degree sequences to improve performance. The paper presents a detailed analysis of the decoding process as a random process on a subgraph of the bipartite graph. They model the progress of the decoding algorithm using differential equations and show that the decoding process can be completed successfully with high probability if the degree sequence of the graph satisfies certain conditions. The authors also discuss practical implementations of the algorithms, including methods for finding good degree sequences and timings of the implementations. The paper concludes with a summary of the main results and recent developments.This paper introduces efficient erasure correcting codes based on sparse bipartite graphs. The authors propose a decoding algorithm for codes derived from cascades of sparse bipartite graphs and analyze it using a discrete-time random process. They derive a criterion for successful decoding based on node degree fractions in the graph. The codes can be encoded in time proportional to ln(1/ε) times the block length n, and can recover from a portion of their entries of length (1+ε)Rn with high probability. The decoding algorithm also runs in time proportional to n ln(1/ε). The algorithms have been implemented and work well in practice. The paper discusses the use of erasure channels, large deviation analysis, and low-density parity-check codes. It introduces a new class of codes that are almost MDS (minimum-distance separable) in the sense that they can recover from a random set of k(1+ε) coordinate places with high probability. These codes have linear time encoding and decoding algorithms and can use an arbitrary alphabet size. The authors describe a construction using bipartite graphs, where each check bit is the XOR of its neighboring message bits. They also discuss cascading codes and the use of irregular degree sequences to improve performance. The paper presents a detailed analysis of the decoding process as a random process on a subgraph of the bipartite graph. They model the progress of the decoding algorithm using differential equations and show that the decoding process can be completed successfully with high probability if the degree sequence of the graph satisfies certain conditions. The authors also discuss practical implementations of the algorithms, including methods for finding good degree sequences and timings of the implementations. The paper concludes with a summary of the main results and recent developments.