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

Physical Layer Network Coding for the K-user Multiple Access Relay Channel

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.Comment: More Simulation results added, 12 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1210.049

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

Distributed Space Time Coding for Wireless Two-way Relaying

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 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 $\mathbb{C}^2$, which are referred to as the singular fade subspaces. DSTC design criterion referred to as the \textit{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.Comment: 27 pages, 4 figures, A mistake in the proof of Proposition 3 given in Appendix B correcte

Physical Layer Network Coding for the Multiple Access Relay Channel

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 $x_A$ of $A$ and $x_B$ of $B,$ and during Phase 2 transmits $f(x_A,x_B),$ 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.Comment: 10 pages, 5 figure

Performance Analysis of Adaptive Physical Layer Network Coding for Wireless Two-way Relaying

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. It was shown 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. Depending on the signal set used at the end nodes, deep fades occur for a finite number of channel fade states referred as the singular fade states. The singular fade states fall into the following two classes: The ones which are caused due to channel outage and whose harmful effect cannot be mitigated by adaptive network coding are referred as the \textit{non-removable singular fade states}. The ones which occur due to the choice of the signal set and whose harmful effects can be removed by a proper choice of the adaptive network coding map are referred as the \textit{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 $\text{SNR}^{-1}$: the error events associated with the removable singular fade states, the error events associated with the non-removable 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 contributing to the average end-to-end SER fall as $\text{SNR}^{-2}$ and as a result the adaptive network coding scheme provides a coding gain over the case when adaptive network coding is not used.Comment: 10 pages, 5 figure