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Random Network Coding over Composite Fields

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Coding Theory and Applications (ICMCTA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10495))

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Abstract

Random network coding is a method that achieves multicast capacity asymptotically for general networks [1, 7]. In this approach, vertices in the network randomly and linearly combine incoming information in a distributed manner before forwarding it through their outgoing edges. To ensure success, the involved finite field needs to be large enough [2, 7], which can be an obstacle if some inner (intermediate) nodes have less computational power than others. In this work, we analyze what can be achieved if different nodes are allowed to use different finite fields from a selection of fields all contained in some composite extension finite field [3, 5].

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Notes

  1. 1.

    Neither this nor the assumption that we have only one sender is really a restriction, as one can always add an artificial vertex \(s^\prime \) to any cycle free network, assume the messages are generated in \(s^\prime \), and for each message add one edge from \(s^\prime \) to the source originally generating it. In this case, the \(a_{i,j}\)’s will need to be chosen in an obvious particular way to reflect the situation.

  2. 2.

    We recall from Sect. 2 that coefficients are set to 0 when they do not correspond to existing connections.

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Acknowledgments

The first listed author gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367). The second listed author acknowledges the support of the TuneSCode project (Grant No. DFF - 1335-00125) granted by the Danish Council for Independent Research and by the Cisco University Research Program Fund (Project CG No. 593761), Gift No. 2015-146035 (3696).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

    Olav Geil

  2. Department of Engineering, Aarhus University, Aarhus, Denmark

    Daniel E. Lucani

Authors
  1. Olav Geil
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  2. Daniel E. Lucani
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Correspondence to Olav Geil .

Editor information

Editors and Affiliations

  1. University of Valladolid, Valladolid, Spain

    Ángela I. Barbero

  2. University of Tartu, Tartu, Estonia

    Vitaly Skachek

  3. Simula@UiB and University of Bergen, Bergen, Norway

    Øyvind Ytrehus

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Geil, O., Lucani, D.E. (2017). Random Network Coding over Composite Fields. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_11

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  • DOI: https://doi.org/10.1007/978-3-319-66278-7_11

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-66278-7

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