## An algebraic approach to network coding (2003)

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Venue: | IEEE/ACM Transactions on Networking |

Citations: | 522 - 86 self |

### BibTeX

@ARTICLE{Koetter03analgebraic,

author = {Ralf Koetter and Muriel Médard and Senior Member},

title = {An algebraic approach to network coding},

journal = {IEEE/ACM Transactions on Networking},

year = {2003},

volume = {11},

pages = {782--795}

}

### Years of Citing Articles

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### Abstract

Abstract—We take a new look at the issue of network capacity. It is shown that network coding is an essential ingredient in achieving the capacity of a network. Building on recent work by Li et al., who examined the network capacity of multicast networks, we extend the network coding framework to arbitrary networks and robust networking. For networks which are restricted to using linear network codes, we find necessary and sufficient conditions for the feasibility of any given set of connections over a given network. We also consider the problem of network recovery for nonergodic link failures. For the multicast setup we prove that there exist coding strategies that provide maximally robust networks and that do not require adaptation of the network interior to the failure pattern in question. The results are derived for both delay-free networks and networks with delays. Index Terms—Algebraic coding, network information theory, network robustness. I.

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Citation Context ...orem in the context of the Ford–Fulkerson algorithm. In Section IV-A, we apply the algebraic framework to acyclic networks. We rapidly recover and extend the work of Li et al. [9] and Ahlswede et al. =-=[10]-=-. In particular, we are able to answer some of the problems left open by the authors [9]. In Section IV-B, we address the general network coding problem for cycle-free networks and we derive necessary... |

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Citation Context ...sidered in the context of networks of links exhibiting ergodic error processes. Channel coding theorems and capacity regions can be found for certain networks of this type, such as broadcast channels =-=[1]-=-–[3], multiple access channels [4], [5], and relay channels [6]–[8]. Recently, some renewed attention has been paid to the capacity of error-free networks. In particular, coding over error-free networ... |

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Citation Context ...error processes. Channel coding theorems and capacity regions can be found for certain networks of this type, such as broadcast channels [1]–[3], multiple access channels [4], [5], and relay channels =-=[6]-=-–[8]. Recently, some renewed attention has been paid to the capacity of error-free networks. In particular, coding over error-free networks for the purpose of transmitting multicast connections has be... |

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Citation Context ...me renewed attention has been paid to the capacity of error-free networks. In particular, coding over error-free networks for the purpose of transmitting multicast connections has been considered [9]–=-=[11]-=-. For a further, recent discussion of network coding, refer to [12, ch. 11, 15]. The work in [9] and [10] examined the network capacity of multicast networks and related capacity to cutsets. Capacity ... |

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Citation Context ...aracterization of this setup is, again, that it is interference free due to the fact that all sinks are supposed to receive all the information. This interference-free situation was also exploited in =-=[15]-=-, where a similar theorem was stated in the context of general, potentially nonlinear, coding strategies. Theorem 8: Let a linear acyclic delay-free network be given with a set of desired connections ... |

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