NDLI: Linear network codes and systems of polynomial equations

Content Provider	IEEE Xplore Digital Library
Author	Dougherty, R. Freiling, C. Zeger, K.
Copyright Year	2008
Description	Author affiliation: Center for Commun. Res., San Diego, CA (Dougherty, R.)
Abstract	If beta and gamma are nonnegative integers and F is a field, then a polynomial collection $\{p_{1},...,$ $p_{beta}\}subeZ[alpha_{1},...,$ $alpha_{gamma}]$ is said to be solvable over F if there exist $omega_{1},...,$ $omega_{gamma}isinF$ such that for all i=1,..., beta we have $p_{i}(omega_{1},...,$ $omega_{gamma})=0.$ We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalar-linear solution over F if and only if the polynomial collection is solvable over F. Koetter and Medardpsilas work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalar-linear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set psi of prime numbers is a set of characteristics of a network if for every qisinpsi, the network has a scalar-linear solution over some finite field with characteristic q and does not have a scalar-linear solution over any finite field whose characteristic lies outside of psi. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co-finite. Two networks N and N' are ls-equivalent if for any finite field F, N is scalar-linearly solvable over F if and only if N' is scalar-linearly solvable over F. We further show that every network is ls-equivalent to a multiple-unicast matroidal network.
Starting Page	1838
Ending Page	1842
File Size	244862
Page Count	5
File Format	PDF
ISBN	9781424422562
DOI	10.1109/ISIT.2008.4595306
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-07-06
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Galois fields Encoding Construction industry Algorithm design and analysis Channel coding Complexity theory
Content Type	Text
Resource Type	Article

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