On the representability of integer polymatroids: Applications in linear code construction
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Medard_On the representability.pdf
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207.85 KB
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Author(s)
Salimi, Amir
Medard, Muriel
Cui, Shuguang
Date Issued
April 2016
Journal
2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Salimi, Amir, Muriel Medard, and Shuguang Cui. “On the Representability of Integer Polymatroids: Applications in Linear Code Construction.” 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), 29 September - October 2, 2015, Monticello, Illinois, IEEE, 2016.
Version
Author's final manuscript
Abstract
It has been shown that there is a duality between the linear network coding solution and the entropic vectors induced by collection of subspaces in a vector space over a finite field (dubbed linearly constructed entropic vectors). The region of all linearly constructed vectors, coincides with the set of all representable polymatroids. For any integer polymatroid, there is an associated matroid, which uniquely identifies the polymatroid. We conjecture that the representability of the underlying matroid is a sufficient condition for integer polymatroids to be linearly representable. We prove that the conjecture holds for representation over real numbers. Furthermore, we show that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
http://dx.doi.org/10.1109/ALLERTON.2015.7447046
