## Efficient erasure correcting codes (2001)

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Venue: | IEEE Transactions on Information Theory |

Citations: | 251 - 20 self |

### BibTeX

@ARTICLE{Luby01efficienterasure,

author = {Michael G. Luby and Michael Mitzenmacher and M. Amin Shokrollahi and Daniel A. Spielman},

title = {Efficient erasure correcting codes},

journal = {IEEE Transactions on Information Theory},

year = {2001},

volume = {47},

pages = {569--584}

}

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### Abstract

Abstract—We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate and any given real number a family of linear codes of rate which can be encoded in time proportional to ��@I A times their block length. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length @IC A or more. The recovery algorithm also runs in time proportional to ��@I A. Our algorithms have been implemented and work well in practice; various implementation issues are discussed. Index Terms—Erasure channel, large deviation analysis, lowdensity parity-check codes. I.

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Citation Context ... it can recover from any set of coordinates of the codeword, i.e., if . Such codes are called minimum-distance separable (MDS) codes. A standard class of MDS codes is given by Reed–Solomon (RS) codes =-=[16]-=-. The connection of these codes with polynomial arithmetic allows for encoding and decoding in time (see, [3, Ch. 11.7] and [16, p. 369]). However, for small values of , quadratic time algorithms are ... |

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Citation Context ...hod was further extended in [21] in which sequences of codes were constructed for which the belief propagation algorithm had a performance extremely close to the Shannon capacity, beating turbo codes =-=[2]-=- by a wide margin for modest block lengths. Another main result of this paper was to show that there are families of degree sequences such that the corresponding graphs asymptotically meet the capacit... |

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Citation Context ...on” algorithm, a numerical procedure to approximatethethresholdofnoisebelowwhichthebeliefpropagationalgorithm5 is asymptotically successful. Several months later, their method was further extended in =-=[21]-=- in which sequences of codes were constructed for which the belief propagation algorithm had a performance extremely close to the Shannon capacity, beating turbo codes [2] by a wide margin for modest ... |

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Citation Context ...erall structure of our codes is related to the low-density parity-check codes introduced by Gallager [6], which have been the subject of a great deal of recent work (see, for example, [10], [11], and =-=[15]-=-). We also use some ideas related to the codes introduced in [25] for error correction. Because we examine the erasure setting, however, our work includes several innovations,s570 IEEE TRANSACTIONS ON... |

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Citation Context ...r programming, and timings of the implementations. In the last section, we summarize the main results of this paper, and discuss recent developments following the publication of a preliminary version =-=[13]-=-. II. GRAPH CODES In this section we introduce a new class of codes. Special subclasses of these codes turn out to be almost MDS in the following sense: an -code in this subclass is capable of recover... |

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Citation Context ... up and extended by other researchers (see, e.g., [14]). Second, the main “concentration theorem” of [10] was extended to a large class of channel models in a landmark paper by Richardson and Urbanke =-=[22]-=-, which first appeared in 1998. Based on their approach, they developed the “density evolution” algorithm, a numerical procedure to approximatethethresholdofnoisebelowwhichthebeliefpropagationalgorith... |

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Citation Context ... general large deviation result which we will prove in Appendix A. Similar results were obtained by Kurtz [8] who studied Markov jump processes, and have been used previously by many researchers, see =-=[5]-=-, [7], [17], [18], [20], [26] and the references therein. We use a version due to Wormald [26] which has the advantage of being directly applicable to our situation. Next we set up the appropriate sys... |

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Citation Context ...ral large deviation result which we will prove in Appendix A. Similar results were obtained by Kurtz [8] who studied Markov jump processes, and have been used previously by many researchers, see [5], =-=[7]-=-, [17], [18], [20], [26] and the references therein. We use a version due to Wormald [26] which has the advantage of being directly applicable to our situation. Next we set up the appropriate system o... |

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Citation Context ...amming approach to design bipartite graphs which give rise to very good codes. Finally, we briefly discuss some of our implementations. A preliminary report on the results of this section appeared in =-=[9]-=-. A. Fraction of Left Nodes Unrecovered So far we have assumed in our analysis that in each layer of the cascade all the check bits are received when trying to recover the message bits. The reason we ... |

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Citation Context ...onal to . The overall structure of our codes is related to the low-density parity-check codes introduced by Gallager [6], which have been the subject of a great deal of recent work (see, for example, =-=[10]-=-, [11], and [15]). We also use some ideas related to the codes introduced in [25] for error correction. Because we examine the erasure setting, however, our work includes several innovations,s570 IEEE... |

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Citation Context ...d decoding algorithms [1] with such a performance guarantee have run times proportional to . The overall structure of our codes is related to the low-density parity-check codes introduced by Gallager =-=[6]-=-, which have been the subject of a great deal of recent work (see, for example, [10], [11], and [15]). We also use some ideas related to the codes introduced in [25] for error correction. Because we e... |

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Citation Context ...nces such that the corresponding graphs asymptotically meet the capacity of the erasure channel (using our simple erasure-decoding algorithm). Another family of such degree sequences was exhibited in =-=[23]-=-. So far, these have been the only known capacity-achieving families of degree sequences, and another example of a communication channel for which capacity-achieving sequences exist for all rates is y... |

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Citation Context ...ed symbols in time proportional to with high probability, where is the block length. These codes can be encoded in time proportional to . The fastest previously known encoding and decoding algorithms =-=[1]-=- with such a performance guarantee have run times proportional to . The overall structure of our codes is related to the low-density parity-check codes introduced by Gallager [6], which have been the ... |

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Citation Context ...d-decisiondecodingalgorithmsgoingbacktoGallager [6]. 4 This paper appears to have been influential. First, the idea of using irregular codes was taken up and extended by other researchers (see, e.g., =-=[14]-=-). Second, the main “concentration theorem” of [10] was extended to a large class of channel models in a landmark paper by Richardson and Urbanke [22], which first appeared in 1998. Based on their app... |

35 |
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Citation Context ... this paper, we assume that the receiver knows the position of each received symbol within the stream of all codeword symbols. We adopt as our model of errors the erasure channel, introduced by Elias =-=[4]-=-, in which each codeword symbol is lost with a fixed constant probability in transit independent of all the other symbols. Elias [4] showed that the capacity of the erasure channel is and that a rando... |

6 |
low-density parity-check codes using irregular graphs
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Citation Context ...o . The overall structure of our codes is related to the low-density parity-check codes introduced by Gallager [6], which have been the subject of a great deal of recent work (see, for example, [10], =-=[11]-=-, and [15]). We also use some ideas related to the codes introduced in [25] for error correction. Because we examine the erasure setting, however, our work includes several innovations,s570 IEEE TRANS... |

1 |
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Citation Context ...milies of degree sequences, and another example of a communication channel for which capacity-achieving sequences exist for all rates is yet to be found. 4 An updated version of this paper appears in =-=[12]-=-. 5 Our erasure decoder turns out to be the belief propagation algorithm for the erasure channel [21].sLUBY et al.: EFFICIENT ERASURE CORRECTING CODES 581 Fig. 8. Length overhead statistics. APPENDIX ... |

1 |
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Citation Context ...eviation result which we will prove in Appendix A. Similar results were obtained by Kurtz [8] who studied Markov jump processes, and have been used previously by many researchers, see [5], [7], [17], =-=[18]-=-, [20], [26] and the references therein. We use a version due to Wormald [26] which has the advantage of being directly applicable to our situation. Next we set up the appropriate system of differenti... |

1 |
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Citation Context ...on result which we will prove in Appendix A. Similar results were obtained by Kurtz [8] who studied Markov jump processes, and have been used previously by many researchers, see [5], [7], [17], [18], =-=[20]-=-, [26] and the references therein. We use a version due to Wormald [26] which has the advantage of being directly applicable to our situation. Next we set up the appropriate system of differential equ... |