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 the vector linear solutions of networks over a field Fq, with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over Fqexists for a network if and only if the network is matroidal with respect to a matroid representable over Fq. We define a discrete polymatroidal network and show that a linear fractional solution over a field Fqexists 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 the networks starting from certain class of discrete polymatroids is provided. Every representation over Fqfor the discrete polymatroid, results in a linear fractional solution over Fqfor the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages X = {x1, x2, . . . xk}, and a set of receivers R, which demand messages. A receiver R ∈ R is specified by the tuple (x, H), where x ∈ X is the message demanded by R and H ⊆ X \ {x} is the side information possessed by R. We first 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. 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. The 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 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.

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 = {x1, x2, ... xk} and a set of receivers R which demand messages. A receiver R ∈ R is specified by the tuple (x,H) where x ∈ X is the message demanded by R and H ⊆ X \ {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.

A linear Fractional Network Coding (FNC) solution over Fqis a linear network coding solution over Fqin 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 Fqand the representability over Fqof 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 Fq, from discrete polymatroids representable over Fq. 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.

We consider the two-user wireless Multiple Access Relay Channel (MARC), in which nodes A and B want to transmit messages to a destination node D with the help of a relay node R. For the MARC, Wang and Giannakis proposed a Complex Field Network Coding (CFNC) scheme. As an alternative, we propose a scheme based on Physical layer Network Coding (PNC), which has so far been studied widely only in the context of two-way relaying. For the proposed PNC scheme, transmission takes place in two phases: (i) Phase 1 during which A and B simultaneously transmit and, R and D receive, (ii) Phase 2 during which A, B and R simultaneously transmit to D. At the end of Phase 1, R decodes the messages xAof A and xBof B, and during Phase 2 transmits f(xA, xB), where f is many-to-one. Communication protocols in which the relay node decodes are prone to loss of diversity order, due to error propagation from the relay node. To counter this, we propose a novel decoder which takes into account the possibility of an error event at R, without having any knowledge about the links from A to R and B to R. It is shown that if certain parameters are chosen properly and if the map f satisfies a condition called exclusive law, the proposed decoder offers the maximum diversity order of two. Also, it is shown that for a proper choice of the parameters, the proposed decoder admits fast decoding, with the same decoding complexity order as that of the CFNC scheme. Simulation results indicate that the proposed PNC scheme performs better than the CFNC scheme.

A Physical layer Network Coding (PNC) scheme is proposed for the K-user wireless Multiple Access Relay Channel (MARC), in which K source nodes transmit their messages to the destination node D with the help of a relay node R. The proposed PNC scheme involves two transmission phases: (i) Phase 1 during which the source nodes transmit, the relay node and the destination node receive and (ii) Phase 2 during which the source nodes and the relay node transmit, and the destination node receives. At the end of Phase 1, the relay node decodes the messages of the source nodes and during Phase 2 transmits a many-to-one function of the decoded messages. Wireless networks in which the relay node decodes, suffer from loss of diversity order if the decoder at the destination is not chosen properly. A novel decoder is proposed for the PNC scheme, which offers the maximum possible diversity order of 2, for a proper choice of certain parameters and the network coding map. Specifically, the network coding map used at the relay is chosen to be a K-dimensional Latin Hypercube, in order to ensure the maximum diversity order of 2. Also, it is shown that the proposed decoder can be implemented by a fast decoding algorithm. Simulation results presented for the 3-user MARC show that the proposed scheme offers a large gain over the existing scheme for the K-user MARC.

The design of modulation schemes for the physical layer network-coded two-way relaying scenario is considered with a protocol which employs two phases: multiple access (MA) phase and broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of MA interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, that this relationship can be used to get the network coding maps satisfying the exclusive law. The channel fade states for which the minimum distance of the effective constellation at the relay become zero are referred to as the singular fade states. For M- PSK modulation ( M any power of 2), it is shown that there are (M2/4- M/2+1 )M singular fade states. Also, it is shown that the constraints which the network coding maps should satisfy so that the harmful effects of the singular fade states are removed, can be viewed equivalently as partially filled Latin Squares (PFLS). The problem of finding all the required maps is reduced to finding a small set of maps for M- PSK constellations ( M any power of 2), obtained by the completion of PFLS. Even though the completability of M ×M PFLS using M symbols is an open problem, specific cases where such a completion is always possible are identified and explicit construction procedures are provided. Having obtained the network coding maps, the set of all possible channel realizations (the complex plane) is quantized into a finite number of regions, with a specific network coding map chosen in a particular region. It is shown that the complex plane can be partitioned into two regions: a region in which any network coding map which satisfies the exclusive law gives the same best performance and a region in which the choice of the network coding map affects the performance. The quantization thus obtained analytically, leads to the same as the one obtained using computer search for 4-PSK signal set by Koike-Akino et al. when specialized for Simulation results show that the proposed scheme performs better than the conventional exclusive-OR (XOR) network coding and in some cases outperforms the scheme proposed by Koike-Akino et al.

The performance analysis of adaptive physical layer network-coded two-way relaying scenario is presented which employs two phases: Multiple access (MA) phase and Broadcast (BC) phase. The deep channel fade conditions which occur at the relay referred as the singular fade states fall in the following two classes: (i) removable and (ii) non-removable singular fade states. With every singular fade state, we associate an error probability that the relay transmits a wrong network-coded symbol during the BC phase. It is shown that adaptive network coding provides a coding gain over fixed network coding, by making the error probabilities associated with the removable singular fade states contributing to the average Symbol Error Rate (SER) fall as SNR-2instead of SNR-1. A high SNR upper-bound on the average end-to-end SER for the adaptive network coding scheme is derived, for a Rician fading scenario, which is found to be tight through simulations. Specifically, it is shown that for the adaptive network coding scheme, the probability that the relay node transmits a wrong network-coded symbol is upper-bounded by twice the average SER of a point-to-point fading channel, at high SNR. Also, it is shown that in a Rician fading scenario, it suffices to remove the effect of only those singular fade states which contribute dominantly to the average SER.

We propose a Physical layer Network Coding (PNC) scheme for the K-user wireless Multiple Access Relay Channel, in which K source nodes want to transmit messages to a destination node D with the help of a relay node R. The proposed scheme involves (i) Phase 1 during which the source nodes alone transmit and (ii) Phase 2 during which the source nodes and the relay node transmit. At the end of Phase 1, the relay node decodes the messages of the source nodes and during Phase 2 transmits a many-to-one function of the decoded messages. To counter the error propagation from the relay node, we propose a novel decoder which takes into account the possibility of error events at R. It is shown that if certain parameters are chosen properly and if the network coding map used at R forms a Latin Hypercube, the proposed decoder offers the maximum diversity order of two. Also, it is shown that for a proper choice of the parameters, the proposed decoder admits fast decoding, with the same decoding complexity order as that of the reference scheme based on Complex Field Network Coding (CFNC). Simulation results indicate that the proposed PNC scheme offers a large gain over the CFNC scheme.

The design of modulation schemes for the physical layer network-coded two-way MIMO relaying scenario is considered, with the denoise-and-forward protocol which employs two phases: Multiple Access phase and Broadcast phase. It is shown that for MIMO two-way relaying, the minimum distance of the effective constellation at the relay becomes zero when all the rows of the channel fade coefficient matrix belong to a finite number of vector subspaces referred to as the singular fade subspaces. The singular fade subspaces can be classified into two kinds based on whether their harmful effects can be removed or not: (i) the removable and (ii) the non-removable singular fade subspaces. It is shown that network coding maps obtained by the completion of appropriate partially filled Latin Rectangles can remove the harmful effects of all the removable singular fade subspaces. For 2λ-PSK signal set, the removable and non-removable singular fade subspaces are characterized and, it is shown that the number of non-removable singular fade subspaces is a small fraction of the total number of singular fade subspaces and this fraction tends to zero as the constellation size tends to infinity. The Latin Rectangles for the case when the end nodes use different number of antennas are shown to be obtainable from the Latin Squares for the case when they use the same number of antennas. Also, the network coding maps which remove all the removable singular singular fade subspaces are shown to be obtainable from a small set of Latin Squares. The removal of all the singular fade subspaces by properly choosing the network coding map, provides a gain of 5.5 dB over the conventional Exclusive-OR network coding, in a Rayleigh fading scenario with 2 antennas at the end nodes and one antenna at the relay node, for 4-PSK signal set.

We consider the wireless two-way relay channel, in which two-way data transfer takes place between the end nodes with the help of a relay. For the Denoise-And-Forward (DNF) protocol, it was shown by Koike-Akino et. al. that adaptively changing the network coding map used at the relay greatly reduces the impact of Multiple Access interference at the relay. The harmful effect of the deep channel fade conditions can be effectively mitigated by a proper choice of these network coding maps at the relay. Alternatively, in this paper we propose a Distributed Space Time Coding (DSTC) scheme, which effectively removes most of the deep fade channel conditions at the transmitting nodes itself without any CSIT and without any need to adaptively change the network coding map used at the relay. It is shown that the deep fades occur when the channel fade coefficient vector falls in a finite number of vector subspaces of ℂ2, which are referred to as the singular fade subspaces. DSTC design criterion referred to as the singularity minimization criterion under which the number of such vector subspaces are minimized is obtained. Also, a criterion to maximize the coding gain of the DSTC is obtained. Explicit low decoding complexity DSTC designs which satisfy the singularity minimization criterion and maximize the coding gain for QAM and PSK signal sets are provided. Simulation results show that at high Signal to Noise Ratio, the DSTC scheme provides large gains when compared to the conventional Exclusive-OR network code and performs slightly better than the adaptive network coding scheme proposed by Koike-Akino et. al.

The design of modulation schemes for the physical layer network-coded two-way MIMO relaying scenario is considered, with the denoise-and-forward (DNF) protocol which employs two phases: Multiple access (MA) phase and Broadcast (BC) phase. It is shown that for MIMO two-way relaying, deep fade occurs at the relay when the row space of the channel fade coefficient matrix is a subspace of a finite number of vector subspaces which are referred to as the singular fade subspaces. It is shown that proper choice of network coding map obtained by the completion of appropriate partially filled Latin Rectangle can remove most of the singular fade subspaces, referred to as the removable singular fade subspaces. For 2λ-PSK signal set, the number of removable and non-removable singular fade subspaces are obtained analytically and it is shown that the number of non-removable singular fade subspaces is a small fraction of the total number of singular fade subspaces. The Latin Rectangles for the case when the end nodes use different number of antennas are shown to be obtainable from the Latin Squares for the case when they use the same number of antennas, irrespective of the number of antennas at the relay. For 2λ-PSK signal set, the singular fade subspaces which are removed by the conventional XOR network code are identified. Also, using the notions of isotopic and transposed Latin Squares, the network coding maps which remove all the removable singular singular fade subspaces are shown to be obtainable from a small set of Latin Squares.

The analysis of modulation schemes for the physical layer network-coded two way relaying scenario is presented which employs two phases: Multiple access (MA) phase and Broadcast (BC) phase. Depending on the signal set used at the end nodes, the minimum distance of the effective constellation seen at the relay becomes zero for a finite number of channel fade states referred as the singular fade states. The singular fade states fall into the following two classes: (i) the ones which are caused due to channel outage and whose harmful effect cannot be mitigated by adaptive network coding called the non-removable singular fade states and (ii) the ones which occur due to the choice of the signal set and whose harmful effects can be removed called the removable singular fade states. In this paper, we derive an upper bound on the average end-to-end Symbol Error Rate (SER), with and without adaptive network coding at the relay, for a Rician fading scenario. It is shown that without adaptive network coding, at high Signal to Noise Ratio (SNR), the contribution to the end-to-end SER comes from the following error events which fall as SNR-1: the error events associated with the removable and nonremovable singular fade states and the error event during the BC phase. In contrast, for the adaptive network coding scheme, the error events associated with the removable singular fade states fall as SNR-2, thereby providing a coding gain over the case when adaptive network coding is not used. Also, it is shown that for a Rician fading channel, the error during the MA phase dominates over the error during the BC phase. Hence, adaptive network coding, which improves the performance during the MA phase provides more gain in a Rician fading scenario than in a Rayleigh fading scenario. Furthermore, it is shown that for large Rician factors, among those removable singular fade states which have the same magnitude, those which have the least absolute value of the phase angle alone contribute dominantly to the end-to-end SER and it is sufficient to remove the effect of only such singular fade states.

The design of modulation schemes for the physical layer network-coded three-way wireless relaying scenario is considered. The protocol employs two phases: Multiple Access (MA) phase and Broadcast (BC) phase with each phase utilizing one channel use. For the two-way relaying scenario, it was observed by Koike-Akino et al. [4], that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called exclusive law. This paper does the equivalent for the three-way relaying scenario. We show that when the three users transmit points from the same 4-PSK constellation, every such network coding map that satisfies the exclusive law can be represented by a Latin Cube of Second Order. The network code map used by the relay for the BC phase is explicitly obtained and is aimed at reducing the effect of interference at the MA stage.

A decode and forward protocol based Trellis Coded Modulation (TCM) scheme for the half-duplex relay channel, in a Rayleigh fading environment, is presented. The proposed scheme can achieve any spectral efficiency greater than or equal to one bit per channel use (bpcu). A near-ML decoder for the suggested TCM scheme is proposed. It is shown that the high Signal to Noise Ratio (SNR) performance of this near-ML decoder approaches the performance of the optimal ML decoder. Based on the derived Pair-wise Error Probability (PEP) bounds, design criteria to maximize the diversity and coding gains are obtained. Simulation results show a large gain in SNR for the proposed TCM scheme over uncoded communication as well as the direct transmission without the relay.

For the design of modulation schemes for the physical layer network-coded two way wireless relaying, it was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA Phase and all these network coding maps should satisfy a requirement called exclusive law. We extend this approach to an Accumulate-Compute and Forward protocol which employs two phases: Multiple Access (MA) phase consisting of two channel uses with independent messages in each channel use, and Broadcast (BC) phase having one channel use. Assuming that the two users transmit points from the same 4-PSK constellation, every such network coding map that satisfies the exclusive law can be represented by a Latin Square with side 16, and conversely, this relationship can be used to get the network coding maps satisfying the exclusive law. Two methods of obtaining this network coding map to be used at the relay are discussed. Using the structural properties of the Latin Squares for a given set of parameters, the problem of finding all the required maps is reduced to finding a small set of maps. Having obtained all the Latin Squares, the set of all possible channel realizations is quantized, depending on which one of the Latin Squares obtained optimizes the performance. The quantization thus obtained, is shown to be the same as the one obtained in [7] for the 2-stage bidirectional relaying.

The design of modulation schemes for the physical layer network-coded two way relaying scenario is considered with the protocol which employs two phases: Multiple access (MA) Phase and Broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA phase. In other words, the set of all possible channel realizations (the complex plane) is quantized into a finite number of regions, with a specific network coding map giving the best performance in a particular region. We obtain such a quantization analytically for the case when M-PSK (for M any power of 2) is the signal set used during the MA phase. We show that the complex plane can be classified into two regions: a region in which any network coding map which satisfies the so called exclusive law gives the same best performance and a region in which the choice of the network coding map affects the performance, which is further quantized based on the choice of the network coding map which optimizes the performance. The quantization thus obtained analytically, leads to the same as the one obtained using computer search for 4-PSK signal set by Koike-Akino et al., for the specific value of M = 4.

The design of modulation schemes for the physical layer network-coded two way relaying scenario is considered with the protocol which employs two phases: Multiple access (MA) Phase and Broadcast (BC) Phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA Phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, and this relationship can be used to get the network coding maps satisfying the exclusive law. Using the structural properties of the Latin Squares for a given set of parameters, the problem of finding all the required maps is reduced to finding a small set of maps for M-PSK constellations. This is achieved using the notions of isotopic and transposed Latin Squares. Furthermore, the channel conditions for which the bit-wise XOR will perform well is analytically obtained which holds for all values of M (for M any power of 2). We illustrate these results for the case where both the end users use QPSK constellation.