Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids

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 $\mathbb{F}_q,$ with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a \textit{discrete polymatroidal} network and show that a linear fractional solution over a field $\mathbb{F}_q,$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb{F}_q$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb{F}_q$ 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 \textit{perfect linear index coding solution} 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.Comment: 24 pages, 6 figures, 4 tables, some sections reorganized, Section VI newly added, accepted for publication in IEEE Transactions on Information Theor

Quasi-linear Network Coding

We present a heuristic for designing vector non-linear network codes for non-multicast networks, which we call quasi-linear network codes. The method presented has two phases: finding an approximate linear network code over the reals, and then quantizing it to a vector non-linear network code using a fixed-point representation. Apart from describing the method, we draw some links between some network parameters and the rate of the resulting code

On the representability of integer polymatroids: Applications in linear code construction

It has been shown that there is a duality between the linear network coding solution and the entropic vectors induced by collection of subspaces in a vector space over a finite field (dubbed linearly constructed entropic vectors). The region of all linearly constructed vectors, coincides with the set of all representable polymatroids. For any integer polymatroid, there is an associated matroid, which uniquely identifies the polymatroid. We conjecture that the representability of the underlying matroid is a sufficient condition for integer polymatroids to be linearly representable. We prove that the conjecture holds for representation over real numbers. Furthermore, we show that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid

On the Fundamental Limits and Symmetric Designs for Distributed Information Systems

Many multi-terminal communication networks, content delivery networks, cache networks, and distributed storage systems can be modeled as a broadcast network. An explicit characterization of the capacity region of the general network coding problem is one of the best known open problems in network information theory. A simple set of bounds that are often used in the literature to show that certain rate tuples are infeasible are based on the graph-theoretic notion of cut. The standard cut-set bounds, however, are known to be loose in general when there are multiple messages to be communicated in the network. This dissertation focuses on broadcast networks, for which the standard cut-set bounds are closely related to union as a specific set operation to combine different simple cuts of the network. A new set of explicit network coding bounds, which combine different simple cuts of the network via a variety of set operations (not just the union), are established via their connections to extremal inequalities for submodular functions. The tightness of these bounds are demonstrated via applications to combination networks. The tightness of generalized cut-set bounds has been further explored by studying the problem of “latency capacity region” for a broadcast channel. An implicit characterization of this region has been proved by Tian, where a rate splitting based scheme was shown to be optimal. However, the explicit characterization of this region was only available when the number of receivers are less than three. In this dissertation, a precise polyhedral description of this region for a symmetric broadcast channel with complete message set and arbitrary number of users has been established. It has been shown that a set of generalized cut-set bounds, characterizes the entire symmetrical multicast region. The achievability part is proved by showing that every maximum rate vector is feasible by using a successive encoding scheme. The framework for achievability strongly relies on polyhedral combinatorics and it can be useful in network information theory problems when a polyhedral description of a region is needed. Moreover, it is known that there is a direct relationship between network coding solution and characterization of entropy region. This dissertation, also studies the symmetric structures in network coding problems and their relation with symmetrical projections of entropy region and introduces new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. Finally, due to the relationship between linear entropic vectors and representability of integer polymatroids, construction of such vector has been discussed. Specifically, It is shown that representability of the particularly constructed matroid is a sufficient condition for integer polymatroids to be linearly representable over real numbers. Furthermore, it has been shown that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid

Linear Index Coding and Representable Discrete Polymatroids

Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections between linear index coding and representable discrete polymatroids. The index coding problem involves a sender which generates a set of messages X = {x(1), x(2), ... x(k)} and a set of receivers R which demand messages. A receiver R is an element of R is specified by the tuple (x, H) where x. X is the message demanded by R and H subset of X \textbackslash {x} is the side information possessed by R. It is first shown 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 perfect linear index coding solution for an index coding problem obtained from that matroid. Multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. 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