Linear Fractional Network Coding and Representable Discrete Polymatroids

A linear Fractional Network Coding (FNC) solution over $\mathbb{F}_q$ is a linear network coding solution over $\mathbb{F}_q$ in which the message dimensions need not necessarily be the same and need not be the same as the edge vector dimension. Scalar linear network coding, vector linear network coding are special cases of linear FNC. In this paper, we establish the connection between the existence of a linear FNC solution for a network over $\mathbb{F}_q$ and the representability over $\mathbb{F}_q$ of discrete polymatroids, which are the multi-set analogue of matroids. All previously known results on the connection between the scalar and vector linear solvability of networks and representations of matroids and discrete polymatroids follow as special cases. An algorithm is provided to construct networks which admit FNC solution over $\mathbb{F}_q,$ from discrete polymatroids representable over $\mathbb{F}_q.$ Example networks constructed from discrete polymatroids using the algorithm are provided, which do not admit any scalar and vector solution, and for which FNC solutions with the message dimensions being different provide a larger throughput than FNC solutions with the message dimensions being equal.Comment: 8 pages, 5 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1301.300

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

Vector Linear Error Correcting Index Codes and Discrete Polymatroids

The connection between index coding and matroid theory have been well studied in the recent past. El Rouayheb et al. established a connection between multi linear representation of matroids and wireless index coding. Muralidharan and Rajan showed that a vector linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Recently index coding with erroneous transmission was considered by Dau et al.. Error correcting index codes in which all receivers are able to correct a fixed number of errors was studied. In this paper we consider a more general scenario in which each receiver is able to correct a desired number of errors, calling such index codes differential error correcting index codes. We show that vector linear differential error correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditionsComment: arXiv admin note: substantial text overlap with arXiv:1501.0506

Guessing Games on Undirected Graphs

PhDGuessing games for directed graphs were introduced by Riis for studying multiple unicast network coding problems. In a guessing game, the players toss generalised die and can see some of the other outcomes depending on the structure of an underlying digraph. They later simultaneously guess the outcome of their own die. Their objective is to find a strategy that maximises the probability that they all guess correctly. The performance of the optimal strategy for a digraph is measured by the guessing number. In general, the existence of an algorithm for computing guessing numbers of a graph is unknown. In the case of undirected graphs, Christofides and Markstr om defined a strategy that they conjectured to be optimal. One of the main results of this thesis is a disproof of this conjecture. In particular, we illustrate an undirected graph on 10 vertices having guessing number which is strictly larger than the lowerbound provided by Christofides and Markstr om's method. Moreover, even in case the undirected graph is triangle-free, we establish counter examples to this conjecture based on combinatorial objects known as Steiner systems. The main tool thus far for computing guessing numbers of graphs has been information theoretic inequalities. Using this method, we are able to derive the guessing numbers of new families of undirected graphs, which in general cannot be computed directly using a computer. A new result of the thesis is that Shannon's information inequalities, which work particularly well for a wide range of graph classes, are not sufficient for computing the guessing number. Another contribution of this thesis is a firm answer to the question concerning irreversible guessing games. In particular, we construct a directed graph G with Shannon upper-bound that is larger than the same bound obtained when we reverse all edges of G. Finally, we initialize a study on noisy guessing game, which is a generalization of noiseless guessing game defined by Riis. We pose a few more interesting questions, some of which we can answer and some which we leave as open problems. 5School of Electronic Engineering and Computer Science

Network and Index Coding with Application to Robust and Secure Communications

Since its introduction in the year 2000 by Ahlswede et al., the network coding paradigm has revolutionized the way we understand information flows in networks. Traditionally, information transmitted in a communication network was treated as a commodity in a transportation network, much like cars on highways or fluids in pipes. This approach, however, fails to capture the very nature of information, which in contrast to material goods, can be coded and decoded. The network coding techniques take full advantage of the inherent properties of information, and allow the nodes in a network, not only to store and forward, but also to "mix", i.e., encode, their received data. This approach was shown to result in a substantial throughput gain over the traditional routing and tree packing techniques. In this dissertation, we study applications of network coding for guarantying reliable and secure information transmission in networks with compromised edges. First, we investigate the construction of robust network codes for achieving network resilience against link failures. We focus on the practical important case of unicast networks with non-uniform edge capacities where a single link can fail at a time. We demonstrate that these networks exhibit unique structural properties when they are minimal, i.e., when they do not contain redundant edges. Based on this structure, we prove that robust linear network codes exist for these networks over GF(2), and devise an efficient algorithm to construct them. Second, we consider the problem of securing a multicast network against an eavesdropper that can intercept the packets on a limited number of network links. We recast this problem as a network generalization of the classical wiretap channel of Type II introduced by Ozarow and Wyner in 1984. In particular, we demonstrate that perfect secrecy can be achieved by using the Ozarow-Wyner scheme of coset coding at the source, on top of the implemented network code. Consequently, we transparently recover important results available in the literature on secure network coding. We also derive new bounds on the required secure code alphabet size and an algorithm for code construction. In the last part of this dissertation, we study the connection between index coding, network coding, and matroid linear representation. We devise a reduction from the index coding problem to the network coding problem, implying that in the linear case these two problems are equivalent. We also present a second reduction from the matroid linear representability problem to index coding, and therefore, to network coding. The latter reduction establishes a strong connection between matroid theory and network coding theory. These two reductions are then used to construct special instances of the index coding problem where vector linear codes outperform scalar linear ones, and where non-linear encoding is needed to achieve the optimal number of transmission. Thereby, we provide a counterexample to a related conjecture in the literature and demonstrate the benefits of vector linear codes

Network coding for robust wireless networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 157-167).Wireless networks and communications promise to allow improved access to services and information, ubiquitous connectivity, and mobility. However, current wireless networks are not well-equipped to meet the high bandwidth and strict delay requirements of future applications. Wireless networks suffer from frequent losses and low throughput. We aim to provide designs for robust wireless networks. This dissertation presents protocols and algorithms that significantly improve wireless network performance and effectively overcome interference, erasures, and attacks. The key idea behind this dissertation is in understanding that wireless networks are fundamentally different from wired networks, and recognizing that directly applying techniques from wired networks to wireless networks limits performance. The key ingredient underlying our algorithms and protocols is network coding. By recognizing the algebraic nature of information, network coding breaks the convention of routing networks, and allows mixing of information in the intermediate nodes and routers. This mixing has been shown to have numerous performance benefits, e.g. increase in throughput and robustness against losses and failures. We present three protocols and algorithms, each using network coding to harness a different characteristic of the wireless medium. We address the problem of interference, erasures, and attacks in wireless networks with the following network coded designs. -- Algebraic NC exploits strategic interference to provide a distributed, randomized code construction for multi-user wireless networks. Network coding framework simplifies the multi-user wireless network model, and allows us to describe the multi-user wireless networks in an algebraic framework. This algebraic framework provides a randomized, distributed code construction, which we show achieves capacity for multicast connections as well as a certain set of non-multicast connections. -- TCP/NC efficiently and reliably delivers data over unreliable lossy wireless networks. TCP, which was designed for reliable transmission over wired networks, often experiences severe performance degradation in wireless networks. TCP/NC combines network coding's erasure correction capabilities with TCP's congestion control mechanism and reliability. We show that TCP/NC achieves significantly higher throughput than TCP in lossy networks; therefore, TCP/NC is well suited for reliable communication in lossy wireless networks. -- Algebraic Watchdog takes advantage of the broadcast nature of wireless networks to provide a secure global self-checking network. Algebraic Watchdog allows nodes to detect malicious behaviors probabilistically, and police their neighbors locally using overheard messages. Unlike traditional detection protocols which are receiver-based, this protocol gives the senders an active role in checking the nodes downstream. We provide a trellis-based inference algorithm and protocol for detection, and analyze its performance. The main contribution of this dissertation is in providing algorithms and designs for robust wireless networks using network coding. We present how network coding can be applied to overcome the challenges of operating in wireless networks. We present both analytical and simulation results to support that network coded designs, if designed with care, can bring forth significant gains, not only in terms of throughput but also in terms of reliability, security, and robustness.by MinJi Kim.Ph.D