NDLI: Insufficiency of linear coding in network information flow

Content Provider	IEEE Xplore Digital Library
Author	Dougherty, R. Freiling, C. Zeger, K.
Copyright Year	2005
Description	Author affiliation: Center for Commun. Res., San Diego, CA (Dougherty, R.)
Abstract	It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution
Starting Page	264
Ending Page	267
File Size	131458
Page Count	4
File Format	PDF
ISBN	0780391519
DOI	10.1109/ISIT.2005.1523335
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2005-09-04
Publisher Place	Australia
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Intelligent networks Galois fields Vectors Mathematics Computer networks Modules (abstract algebra) Network coding Linearity Convolutional codes Filter bank
Content Type	Text
Resource Type	Article

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