Network Coding is a technique which looks beyond traditional store-and-forward approach followed by routers and switches in communication networks, and as an extension introduces maps termed as ‘local encoding kernels’ and ‘global encoding kernels’ defined for each communication link in the network. The purpose of both these maps is to define rules as to how to combine the packets input on the node to form a packet going out on an edge. The paradigm of network coding was formally and for the first time introduced by Ahlswede et al. in [1], where they also demonstrated its use in case of single-source multiple-sink network multicast, although with use of much complex mathematical apparatus. In [1], examples of networks are also presented where it is shown that network coding can improve the overall throughput of the network which can not otherwise be realized by the conventional store-and-forward approach. The main result in [1], i.e. the capacity of single-source multiple-sinks information network is nothing but the minimum of the max-flows from source to each sink, was again proved by Li, Yeung, and Cai in [2] where they showed that only linear operations suffice to achieve the capacity of multicast network. The authors in [2] defined generalizations to the multicast problem, which they termed as linear broadcast, linear dispersion, and Generic LCM as strict generalizations of linear multicast, and showed how to build linear network codes for each of these cases. For the case of linear multicast, Koetter and Medard in [3] developed an algebraic framework using tools from algebraic geometry which also proved the multicast max-flow min-cut theorem proved in [1] and [2]. It was shown that if the size of the finite field is bigger than a certain threshold, then there always exists a solution to the linear multicast, provided it is solvable. In other words, a solvable linear multicast always has a solution in any finite field whose cardinality is greater than the threshold value. The framework in [3] also dealt with the general linear network coding problem involving multiple sources and multiple sinks with non-uniform demand functions at the sinks, but did not touched upon the key problem of finding the characteristic(s) of the field in which it may have solution. It was noted in [5] that a solvable network may not have a linear solution at all, and then introduced the notion of general linear network coding, where the authors conjectured that every solvable network must have a general linear solution. This was refuted by Dougherty, Freiling, Zeger in [6], where the authors explicitly constructed example of a solvable network which has no general linear solution, and also networks which have solution in a finite field of char 2, and not in any other finite field. But an algorithm to find the characteristic of the field in which a scalar or general linear solution(if at all) exists did not find any mention in [3] or [6]. It was a simultaneous discovery by us(as part of this thesis) as well as by Dougherty, Freiling, Zeger in [7] to determine the characteristics algorithmically. Applications of Network Coding techniques to wireless networks are seen in literature( [8], [9], [10]), where [8] provided a variant of max-flow min-cut theorem for wireless networks in the form of linear programming constraints. A new architecture termed as COPE was introduced in [10] which used opportunistic listening and opportunistic coding in wireless mesh networks.

The network coding is a new paradigm that has been shown to improve throughput, fault tolerance, and other quality of service parameters in communication networks. The basic idea of the network coding techniques is to relish the "mixing" nature of the information flows, i.e., many algebraic operations (e.g., addition, subtraction etc.) can be performed over the data packets. Whereas traditionally information flows are treated as physical commodities (e.g., cars) over which algebraic operations can not be performed. In this dissertation we answer some of the important open questions related to the network coding. Our work can be divided into four major parts. Firstly, we focus on network code design for the dynamic networks, i.e., the networks with frequently changing topologies and frequently changing sets of users. Examples of such dynamic networks are content distribution networks, peer-to-peer networks, and mobile wireless networks. A change in the network might result in infeasibility of the previously assigned feasible network code, i.e., all the users might not be able to receive their demands. The central problem in the design of a feasible network code is to assign local encoding coefficients for each pair of links in a way that allows every user to decode the required packets. We analyze the problem of maintaining the feasibility of a network code, and provide bounds on the number of modifications required under dynamic settings. We also present distributed algorithms for the network code design, and propose a new path-based assignment of encoding coefficients to construct a feasible network code. Secondly, we investigate the network coding problems in wireless networks. It has been shown that network coding techniques can significantly increase the overall throughput of wireless networks by taking advantage of their broadcast nature. In wireless networks each packet transmitted by a device is broadcasted within a certain area and can be overheard by the neighboring devices. When a device needs to transmit packets, it employs the Index Coding that uses the knowledge of what the device's neighbors have heard in order to reduce the number of transmissions. With the Index Coding, each transmitted packet can be a linear combination of the original packets. The Index Coding problem has been proven to be NP-hard, and NP-hard to approximate. We propose an efficient exact, and several heuristic solutions for the Index Coding problem. Noting that the Index Coding problem is NP-hard to approximate, we look at it from a novel perspective and define the Complementary Index Coding problem, where the objective is to maximize the number of transmissions that are saved by employing coding compared to the solution that does not involve coding. We prove that the Complementary Index Coding problem can be approximated in several cases of practical importance. We investigate both the multiple unicast and multiple multicast scenarios for the Complementary Index Coding problem for computational complexity, and provide polynomial time approximation algorithms. Thirdly, we consider the problem of accessing large data files stored at multiple locations across a content distribution, peer-to-peer, or massive storage network. Parts of the data can be stored in either original form, or encoded form at multiple network locations. Clients access the parts of the data through simultaneous downloads from several servers across the network. For each link used client has to pay some cost. A client might not be able to access a subset of servers simultaneously due to network restrictions e.g., congestion etc. Furthermore, a subset of the servers might contain correlated data, and accessing such a subset might not increase amount of information at the client. We present a novel efficient polynomial-time solution for this problem that leverages the matroid theory. Fourthly, we explore applications of the network coding for congestion mitigation and over flow avoidance in the global routing stage of Very Large Scale Integration (VLSI) physical design. Smaller and smarter devices have resulted in a significant increase in the density of on-chip components, which has given rise to congestion and over flow as critical issues in on-chip networks. We present novel techniques and algorithms for reducing congestion and minimizing over flows.

Network coding has emerged as an attractive alternative to routing because of the through put improvement it provides by reducing the number of channel uses. In a wireless scenario, in addition, further improvement can be obtained through Physical layer Network Coding (PNC), a technique in which nodes are allowed to transmit simultaneously, instead of transmitting in orthogonal slots. In this thesis, the design and analysis of network coding schemes are considered, for wireless two-way relaying, multi-user Multiple Access Relay Channel (MARC) and wireline networks. In a wireless two-way relay channel with PNC, the simultaneous transmissions of user nodes result in Multiple Access Interference (MAI) at there lay node. The harmful effect of MAI is the presence of signal set dependent deep channel fade conditions, called singular fade states, under which the minimum distance of the effective constellation at the relay become zero. Adaptively changing the network coding map used at the relay according to channel conditions greatly reduces the impact of this MAI. In this work, we obtain these adaptive PNC maps, which are finite in number ,by completing partially filled Latin Squares and using graph vertex coloring. Having obtained the network coding maps, the set of all possible channel realizations is quantized into a finite number of regions, with a specific network coding map chosen in a particular region and such a quantization is obtained analytically for 2λ-PSK signal set. The performance of the adaptive PNC scheme for two-way relaying is analyzed and tight high SNR upper bounds are obtained for the average end-to-end symbol error probability, in terms of the average error probability of a point-to-point fading channel. The adaptive PNC scheme is generalized for two-way relaying with multiple antennas at the nodes. As an alternative to the adaptive PNC scheme for two-way relaying, a Distributed Space Time Coding (DSTC) scheme is proposed, which effectively re-moves the effect of singular fade states at the transmitting nodes itself without any Channel State Information at the Transmitter (CSIT), and without any need to change the PNC map as a function of channel fade conditions. It is shown that the singular fade states can be viewed equivalently as vector subspaces of C2, which are referred to as the singular fade subspaces. DSTC design criterion to minimize the number of singular fade subspaces and maximize the coding gain is formulated and explicit low decoding complexity DSTC designs are provided. For the K-user MARC, in which K source nodes want to transmit messages to a destination node D with the help of are lay node R, a new PNC scheme is proposed. Use of a many-to-one PNC map with conventional minimum squared Euclidean distance decoding at D, results in a loss of diversity order due to error propagation from the relay node. To counter this, we propose a novel low complexity decoder which offers the maximum diversity order of two. Next, we consider wire line networks and explore the connections between linear network coding, linear index coding and discrete polymatroids, which are the multi-set analogue of matroids. We define a discrete polymatroidal network and show that a fractional vector linear solution over a field Fq exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over Fq.An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over Fq for the discrete polymatroid, results in a fractional vector linear solution over Fq for the constructed network. It is 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.