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.

Multiple users compete for a common resource like bandwidth to communicate data in interference networks. Existing approaches in dealing with interference limit the rate of communication due to paucity of shared resources. This limitation in the rate gets more glaring as the number of users in the network increases. For example, existing wireless systems either choose to orthogonalize the users (for example, Frequency Division Multiple Access (FDMA) systems or Code Division Multiple Access (CDMA) systems) or treat interference as Gaussian noise at the receivers. It is well known that these approaches are sub-optimal in general. Orthogonalization of users limit the number of available interference-free channels (known as degrees of freedom, abbreviated as DoF) and treating interference as noise means that the receiver cannot make use of the structure in the interfering signals. This motivates the need to analyze alternate transmit and decoding schemes in interference networks. This thesis mainly analyzes transmit schemes that use linear precoding for various configurations of interference networks with some practical constraints imposed by the use of finite input constellations, propagation delays, and channel state availability at the transmitters. The main contributions of this thesis are listed below. Achievable rates using precoding with finite constellation inputs in Gaussian Interference Channels (GIC) is analyzed. A metric for finding the approximate angle of rotation to maximally enlarge the Constellation Constrained (CC) capacity of two-user Gaussian Strong Interference Channel (GSIC) is proposed. Even as the Gaussian alphabet FDMA rate curve touches the capacity curve of the GSIC, with both the users using the same finite constellation, we show that the CC FDMA rate curve lies strictly inside the CC capacity curve at high powers. For a K-user MIMO GIC, a set of necessary and sufficient conditions on the precoders under which the mutual information between between relevant transmit-receive pairs saturate like in the single user case is derived. Gradient-ascent based algorithms to optimize the sum-rate achieved by precoding with finite constellation inputs and treating interference as noise are proposed. For a class of Gaussian interference networks with general message demands, identified as symmetrically connected interference networks, the expected sumspectral efficiency (in bits/sec/Hz) is shown to grow linearly with the number of transmitters at finite SNR, using a time-domain Interference Alignment (IA) scheme in the presence of line of sight (LOS) channels. For a 2×2 MIMO X-Network with M antennas at each node, we identify spacetime block codes that could be coupled with an appropriate precoding scheme to achieve the maximum possible sum-DoF of 4M 3 , for M = 3, 4. The proposed schemes are shown to achieve a diversity gain of M with SNR-independent finite constellation inputs. The proposed schemes have lower CSIT requirements compared to existing schemes. This thesis also makes an attempt to guarantee a minimum throughput when the zero-interference conditions cannot be satisfied in a wireline network with three unicast sessions with delays, using Precoding Based Network Alignment (PBNA). Three different PBNA schemes namely PBNA with time-varying local encoding coefficients (LECs), PBNA using transform approach and time-invariant LECs, and PBNA using transform approach and block time-varying LECs are proposed and their feasibility conditions analyzed.

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.