Networks, Matroids, and Non-Shannon Information Inequalities

Dougherty, Randall; Freiling, Chris; Zeger, K.

doi:10.1109/tit.2007.896862

Cited by 215 publications

(305 citation statements)

References 25 publications

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“…Let M = (S, I) be a matroid with rank function ρ. The network N is a matroidal network (see [7]) associated with M if there exists a function f : µ ∪ → S such that the following conditions hold:…”

Section: Network Construction From Polynomial Systemmentioning

confidence: 99%

“…It was shown in [7] that many interesting networks are matroidal, including all networks that are scalar-linearly solvable over a finite field (e.g. solvable multicast networks).…”

Section: Network Construction From Polynomial Systemmentioning

confidence: 99%

“…In [7], a method was presented for constructing, from given matroids, (matroidal) networks which reflect some of the matroids' properties. This construction was used to obtain networks used to prove various results in the literature [5], [6], [8].…”

Section: Network Construction From Polynomial Systemmentioning

confidence: 99%

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Linear Network Codes and Systems of Polynomial Equations

Dougherty

Freiling

Zeger

2008

IEEE Trans. Inform. Theory

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Abstract-If β and γ are nonnegative integers and F is a field, then a polynomial collection {p1, . . . , p β } ⊆ Z[α1, . . . , αγ ] is said to be solvable over F if there exist ω1, . . . , ωγ ∈ F such that for all i = 1, . . . , β we have pi(ω1, . . . , ωγ ) = 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 Médard's 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 Ψ of prime numbers is a set of characteristics of a network if for every q ∈ Ψ, the network has a scalarlinear 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 Ψ. 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.

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Section: Network Construction From Polynomial Systemmentioning

confidence: 99%

“…It was shown in [7] that many interesting networks are matroidal, including all networks that are scalar-linearly solvable over a finite field (e.g. solvable multicast networks).…”

Section: Network Construction From Polynomial Systemmentioning

confidence: 99%

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Linear Network Codes and Systems of Polynomial Equations

Dougherty

Freiling

Zeger

2008

IEEE Trans. Inform. Theory

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“…Notice that vector operations imply linearity over vectors; therefore, a vector solution is always a (vector) linear solution. In terms of the achievable rate, vector network coding outperforms scalar linear network coding [10], [35]. In [35], an example of a network which is not scalar linear solvable but is solvable by vector routing was shown.…”

Section: Introductionmentioning

confidence: 99%

Vector network coding based on subspace codes outperforms scalar linear network coding

Etzion

Wachter-Zeh

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-This paper considers vector network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network. The multicast networks considered in this paper have one source with h messages, and the vector solution is over a field of size q with vectors of length t. For a given network, let the smallest field size for which the network has a scalar linear solution be qs, then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be qs −q t . In this contribution, the achieved gap is q (h−2)t 2 /h+o(t) for any q ≥ 2 and any even h ≥ 4. If h ≥ 5 is odd, then the achieved gap of the alphabet size is q (h−3)t 2 /(h−1)+o(t) . Previously, only a gap of size size one had been shown for networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks. For three messages, we also show an advantage of vector network coding, while for two messages the problem remains open. Several networks are considered, all of them are generalizations and modifications of the well-known combination networks. The vector network codes that are used as solutions for those networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

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“…Although these non-shannon type inequalities give outer bounds for F4, its complete characterization remains an open problem. The non-Shannon inequalities have also proven useful in deriving various outerbounds for different network information theory problems [10], [11]. However, there has been much less focus on determining inner bounds on Fn [4], [12].…”

Section: Introductionmentioning

confidence: 99%

On a Construction of Entropic Vectors Using Lattice-Generated Distributions

Hassibi

Shadbakht

2007

2007 IEEE International Symposium on Information Theory

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Abstract-The problem of determining the region of entropic vectors is a central one in information theory. Recently, there has been a great deal of interest in the development of nonShannon information inequalities, which provide outer bounds to the aforementioned region; however, there has been less recent work on developing inner bounds. This paper develops an inner bound that applies to any number of random variables and which is tight for 2 and 3 random variables (the only cases where the entropy region is known). The construction is based on probability distributions generated by a lattice. The region is shown to be a polytope generated by a set of linear inequalities. Study of the region for 4 and more random variables is currently under investigation.

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Networks, Matroids, and Non-Shannon Information Inequalities

Cited by 215 publications

References 25 publications

Linear Network Codes and Systems of Polynomial Equations

Linear Network Codes and Systems of Polynomial Equations

Vector network coding based on subspace codes outperforms scalar linear network coding

On a Construction of Entropic Vectors Using Lattice-Generated Distributions

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