## A linear time erasure-resilient code with nearly optimal recovery (1996)

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Venue: | IEEE Transactions on Information Theory |

Citations: | 47 - 6 self |

### BibTeX

@ARTICLE{Alon96alinear,

author = {Noga Alon and Michael Luby},

title = {A linear time erasure-resilient code with nearly optimal recovery},

journal = {IEEE Transactions on Information Theory},

year = {1996},

volume = {42},

pages = {1732--1736}

}

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### Abstract

We develop an efficient scheme that produces an encoding of a given message such that the message can be decoded from any portion of the encoding that is approximately equal to the length of the message. More precisely, an (n, c, ℓ, r)-erasure-resilient code consists of an encoding algorithm and a decoding algorithm with the following properties. The encoding algorithm produces a set of ℓ-bit packets of total length cn from an n-bit message. The decoding algorithm is able to recover the message from any set of packets whose total length is r, i.e., from any set of r/ℓ packets. We describe erasure-resilient codes where both the encoding and decoding algorithms run in linear time and where r is only slightly larger than n. 1

### Citations

2114 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ...ompetitive in practice with the simpler quadratic methods except for extremely large messages, and even then, their time complexity is at least\Omega\Gamma n log n), see, e.g., [10], Chapter 11.7 and =-=[13]-=-, p. 369. Thus, the design of highly efficient algorithms for implementing erasure-resilient codes is interesting theoretically and important for practical applications. Our scheme has the property th... |

1728 | The Probabilistic Method
- Alon, Spencer
- 2000
(Show Context)
Citation Context ... into packet Pw j . Let I be any subset of V with jIj = (1+ffl)n c 0 ` . For each i 2 V , let d i denote the number of letters of E i that are in the packets indexed by I. By a lemma in [6] (see also =-=[7]-=-, page 122), X i2V (d i \Gamma jIjc 0 ` 2 =N) 2 2 jIj(1 \Gamma jIj`=N)s4c 0 `jIjs8n: (1) Note that jIjc 0 ` 2 N = (1 + ffl)n` N = (1 + ffl)` 1 + fls(1 + fl)`: (2) Let M be the set of i 2 V for which t... |

491 | Efficient Dispersal of Information for Security, Load Balancing and Fault Tolerance
- Rabin
- 1997
(Show Context)
Citation Context ...this approach have been developed for applications such as multicasting real-time high-volume video information over lossy packet based networks [3, 2, 9] and other high volume real-time applications =-=[14]-=-. The two most important properties of erasure-resilient codes in these applications are the running times of the encoding and decoding algorithms and the amount of encoding sufficient to recover the ... |

450 |
Theory and Practice of Error Control Codes
- Blahut
- 1984
(Show Context)
Citation Context ...these methods are not competitive in practice with the simpler quadratic methods except for extremely large messages, and even then, their time complexity is at least\Omega\Gamma n log n), see, e.g., =-=[10]-=-, Chapter 11.7 and [13], p. 369. Thus, the design of highly efficient algorithms for implementing erasure-resilient codes is interesting theoretically and important for practical applications. Our sch... |

290 | Expander codes
- Sipser, Spielman
- 1996
(Show Context)
Citation Context ...ms, then so does the resulting erasure-resilient code. The recent breakthrough result of Spielman [16] on error correcting codes is directly relevant to our scheme. Spielman applies the techniques in =-=[15]-=- and [8], and constructs linear time error correcting codes with linear rate and linear minimum distance. This error correcting code stretches an nbit message to a cn-bit message and can recover the m... |

283 | Priority encoding transmission
- Albanese, Blomer, et al.
- 1996
(Show Context)
Citation Context ...ncy using erasure-resilient codes. Algorithms based on this approach have been developed for applications such as multicasting real-time high-volume video information over lossy packet based networks =-=[3, 2, 9]-=- and other high volume real-time applications [14]. The two most important properties of erasure-resilient codes in these applications are the running times of the encoding and decoding algorithms and... |

218 |
Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators
- Margulis
- 1988
(Show Context)
Citation Context ...here that there are sufficiently many expanders in these families whose number of nodes is divisible by any desired constant. It is not difficult to show that this assumption can be omitted. By [11], =-=[12]-=- the sequence of integers m for which there is a (d; 2 p d \Gamma 1)- expander on m nodes is a dense sequence. We need the following from [6]. Proposition 1 [6] The number of edges induced by any set ... |

122 |
Performance evaluation of forward error correction
- Biersack
- 1992
(Show Context)
Citation Context ...ncy using erasure-resilient codes. Algorithms based on this approach have been developed for applications such as multicasting real-time high-volume video information over lossy packet based networks =-=[3, 2, 9]-=- and other high volume real-time applications [14]. The two most important properties of erasure-resilient codes in these applications are the running times of the encoding and decoding algorithms and... |

122 | Linear-time encodeable and decodable error-correcting codes
- Spielman
- 1996
(Show Context)
Citation Context ...set. Moreover, if the corresponding error correcting code has efficient encoding and decoding algorithms, then so does the resulting erasure-resilient code. The recent breakthrough result of Spielman =-=[16]-=- on error correcting codes is directly relevant to our scheme. Spielman applies the techniques in [15] and [8], and constructs linear time error correcting codes with linear rate and linear minimum di... |

117 | Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
- Alon, Bruck, et al.
- 1992
(Show Context)
Citation Context ... is missing. Our scheme is based on the properties of expanders which are explicit 3 graphs with pseudo-random properties. The relevance of these graphs to error correcting codes has been observed in =-=[5]-=-, and indeed we apply some of the ideas of that paper. Erasure-resilient codes are related to error correcting codes, and are typically easier to design. For example, an error correcting code with enc... |

107 | Explicit construction of linear sized tolerant networks
- Alon, Chung
- 1986
(Show Context)
Citation Context ...ult to show that this assumption can be omitted. By [11], [12] the sequence of integers m for which there is a (d; 2 p d \Gamma 1)- expander on m nodes is a dense sequence. We need the following from =-=[6]-=-. Proposition 1 [6] The number of edges induced by any set of x nodes in a (d; )-graph on m nodes does not exceed 1 2 x(d x m + (1 \Gamma x m )): 6 Proof of Lemma 1: Fix an integer d, where 64 fl 02 !... |

100 |
Deterministic simulation in LOGSPACE
- Ajtai, Komlós, et al.
- 1987
(Show Context)
Citation Context ...hods in places where we use quadratic time MDS codes will be as efficient in practice. The construction in Section 4 can be improved by using walks in expanders instead of edges, using the methods of =-=[1]-=-. The relevance of this method to the case of expander based error correcting codes has been observed by us (cf. [15]), and a similar remark holds here also. Combining our technique here with the meth... |

71 |
Explicit Expanders and the Ramanujan Conjectures
- Lubotzky, Phillips, et al.
- 1986
(Show Context)
Citation Context ...ssume here that there are sufficiently many expanders in these families whose number of nodes is divisible by any desired constant. It is not difficult to show that this assumption can be omitted. By =-=[11]-=-, [12] the sequence of integers m for which there is a (d; 2 p d \Gamma 1)- expander on m nodes is a dense sequence. We need the following from [6]. Proposition 1 [6] The number of edges induced by an... |

34 | Linear time erasure codes with nearly optimal recovery
- Alon, Edmonds, et al.
(Show Context)
Citation Context ...ent code obtained by a deterministic scheme. A preliminary description of this scheme, together with descriptions of probabilistic schemes that are more efficient based on this scheme can be found in =-=[4]-=-. The scheme has the property that, on inputs n, c, and ffl, the run time of the (n; c)-code is O(n=ffl 4 ), it is (1 + ffl)-MDS, and the packet size is O((1=ffl 4 ) log(1=ffl)). Note that for constan... |

3 |
Problems of complexity in the theory of correcting codes
- Bassalygo, Zyablov, et al.
- 1977
(Show Context)
Citation Context ...so does the resulting erasure-resilient code. The recent breakthrough result of Spielman [16] on error correcting codes is directly relevant to our scheme. Spielman applies the techniques in [15] and =-=[8]-=-, and constructs linear time error correcting codes with linear rate and linear minimum distance. This error correcting code stretches an nbit message to a cn-bit message and can recover the message w... |

2 |
Priority Encoding Transmission", ICSI
- Albanese, Blomer, et al.
- 1994
(Show Context)
Citation Context ...ncy using erasure-resilient codes. Algorithms based on this approach have been developed for applications such as multicasting real-time high-volume video information over lossy packet based networks =-=[3, 2, 9]-=- and other high volume real-time applications [14]. The two most important properties of erasure-resilient codes in these applications are the running times of the encoding and decoding algorithms and... |

2 |
Priority Encoding Transmission”, ICSI
- Albanese, Blömer, et al.
- 1994
(Show Context)
Citation Context |