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 @math with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a network and show that a linear fractional solution over a field @math exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over @math An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over @math for the discrete polymatroid, results in a linear fractional solution over @math 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 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.
In scalar and vector network coding, it is inherently assumed that the dimensions of the message vectors are the same and also the same as the dimensions of the vectors carried in the edges of the network. It is possible that a network does not admit any scalar or vector solution, but admits a solution if all the dimensions of the message vectors are not equal to the edge vector dimension. Such network coding solutions, called Fractional Network Coding (FNC) solutions have been considered in @cite_12 @cite_9 @cite_4 . The work in @cite_12 primarily focusses on fractional routing, which is a special case of FNC. In @cite_9 , algorithms were provided to compute the capacity region for a network, which was defined to be the closure of all rates achievable using FNC. In @cite_4 , achievable rate regions for certain specific networks were found and it was shown that achievable rate regions using linear FNC need not be convex.
@cite_9 @cite_4 @cite_12
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Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popular conjecture on the benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist (singlesource) multicast networks that are vector linearly solvable of dimension L over GF(q) but not scalar linearly solvable over any field of size q' qL. This paper introduces a systematic way to construct such multicast networks, and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes pL with respect to an arbitrary prime p. On the other hand, this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension L over GF(2) but have scalar linear solutions over GF(q') for someq' <; 2 L , where q' can be odd or even. This discovery also unveils that over a given base field, a multicast network that has a vector linear solution of dimension L does not necessarily have a vector linear solution of dimension L' > L. We investigate a channel-coded physical-layer network coding (CPNC) scheme for binary-input Gaussian two-way relay channels. In this scheme, the codewords of the two users are transmitted simultaneously. The relay computes and forwards a network-coded (NC) codeword without complete decoding of the two users' individual messages. We propose a new punctured codebook method to explicitly find the distance spectrum of the CPNC scheme. Based on that, we derive an asymptotically tight performance bound for the error probability. Our analysis shows that, compared to the single-user scenario, the CPNC scheme exhibits the same minimum Euclidean distance but an increased multiplicity of error events with minimum distance. At a high SNR, this leads to an SNR penalty of at most ln2 (in linear scale), for long channel codes of various rates. Our analytical results match well with the simulated performance. Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern. Consider a network of @math n processes, where each process inputs a @math d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single @math d-dimensional vector @math v?Rd, inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple @math d-dimensional vectors in @math Rd, all within a fixed Euclidean distance @math ∈ of each other, and inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to @math f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, @math n>max 3f,(d+1)f is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, @math n>(d+2)f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.