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**1 - 2**of**2**### 1A Matroid Theory Approach to Multicast Network Coding

"... Abstractâ€”Network coding encourages the mixing of informa-tion flows at intermediate nodes of a network for enhanced network capacity, especially for one-to-many multicast applica-tions. A fundamental problem in multicast network coding is to construct a feasible solution such that encoding and decod ..."

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Abstractâ€”Network coding encourages the mixing of informa-tion flows at intermediate nodes of a network for enhanced network capacity, especially for one-to-many multicast applica-tions. A fundamental problem in multicast network coding is to construct a feasible solution such that encoding and decoding are performed over a finite field of size as small as possible. Coding operations over very small finite fields (e.g., F2) enable low computational complexity in theory and ease of implementation in practice. In this work, we propose a new approach based on matroid theory to study multicast network coding and its minimum field size requirements. Applying this new approach that translates multicast networks into matroids, we derive the first upper-bounds on the field size requirement based on the number of relay nodes in the network, and make new progresses along the direction of proving that coding over very small fields (F2 and F3) suffices for multicast network coding in planar networks. I.

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, 2013

"... Network Coding encourages information coding across a communication network. While the ne-cessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on al ..."

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Network Coding encourages information coding across a communication network. While the ne-cessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an in-depth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a meta-conjecture, the NC-Minor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NC-Minor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and KO(q / log q) minors, for networks requiring F3, F4, F5 and Fq, respectively. We finally prove that network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed.