Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of networks over a field @math with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a network and show that a linear fractional solution over a field @math exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over @math An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over @math for the discrete polymatroid, results in a linear fractional solution over @math for the constructed network. Next, we consider the index coding problem and show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a for an index coding problem obtained from that matroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.
The concept of network coding, originally introduced by Ahlswede et. al. in @cite_16 , helps towards providing more throughput in a communication network than what pure routing solutions provide. For multicast networks, it was shown in @cite_6 that linear solutions exist for sufficiently large field size. An algebraic framework for finding linear solutions in networks was introduced in @cite_7 .
@cite_16 @cite_7 @cite_6
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Network coding shows that the data rate can be increased if information is allowed to be encoded in the network nodes. The recent work of S.-Y.R. (see IEEE Trans. Inform. Theory, vol.49, no.2, p.371-81, 2003) and R. Koetter and M. Medard (see Proc. INFOCOM, 2002) shows that linear network coding is sufficient for a single source multicast network. The restrictiveness of the use of linear code is still an unknown in the general multisource multicast network. We characterize the achievable information rate region for a single source node multi-source multicast network with two sinks. We further show that linear coding is sufficient for achieving the maximum network capacity. We present a distributed random linear network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves capacity with probability exponentially approaching 1 with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error exponents for linear Slepian-Wolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two examples of practical scenarios: distributed network operation and networks with dynamically varying connections. Our derivation of these results also yields a new bound on required field size for centralized network coding on general multicast networks Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node. We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.