We consider communication over a noisy network under randomized linear network coding. Possible error mechanism include node- or link- failures, Byzantine behavior of nodes, or an over-estimate of the network min-cut. Building on the work of Kotter and Kschischang, we introduce a probabilistic model for errors. We compute the capacity of this channel and we define an error-correction scheme based on random sparse graphs and a low-complexity decoding algorithm. By optimizing over the code degree profile, we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability.
The most common scenario studied in this context is a multi-cast one in which the source aims at communicating the same information to a set of receivers (distinct nodes in the same network.) The fundamental theorem of network coding states that this is possible using network coding (i.e., processing at the nodes) if the values of the min-cuts from the source to any of the receivers is at least @math @cite_4 . Moreover, linear network coding suffices @cite_9 . This means that processing at the nodes can be limited to forwarding packets which are linear (over @math ) combinations of incoming packets. Finally, it is not necessary to choose the local encoding functions at the nodes carefully. Random linear combinations are sufficient with probability close to one, provided the cardinality of the field is large enough @cite_13 @cite_0 . For a general introduction into network coding we refer the reader to @cite_14 @cite_10 @cite_5 .
@cite_14 @cite_4 @cite_10 @cite_9 @cite_0 @cite_5 @cite_13
2123562143 2014414302 2139416652 2950479988
The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures. This paper studies the zero-delay source-channel coding problem, and specifically the problem of obtaining the vector transformations that optimally map between the m-dimensional source space and k-dimensional channel space, under a given transmission power constraint and for the mean square error distortion. The functional properties of the cost are studied and the necessary conditions for the optimality of the encoder and decoder mappings are derived. An optimization algorithm that imposes these conditions iteratively, in conjunction with the noisy channel relaxation method to mitigate poor local minima, is proposed. The numerical results show strict improvement over prior methods. The numerical approach is extended to the scenario of source-channel coding with decoder side information. The resulting encoding mappings are shown to be continuous relatives of, and in fact subsume as special case, the Wyner-Ziv mappings encountered in digital distributed source coding systems. A well-known result in information theory pertains to the linearity of optimal encoding and decoding mappings in the scalar Gaussian source and channel setting, at all channel signal-to-noise ratios (CSNRs). In this paper, the linearity of optimal coding, beyond the Gaussian source and channel, is considered and the necessary and sufficient condition for linearity of optimal mappings, given a noise (or source) distribution, and a specified a total power constraint are derived. It is shown that the Gaussian source-channel pair is unique in the sense that it is the only source-channel pair for which the optimal mappings are linear at more than one CSNR values. Moreover, the asymptotic linearity of optimal mappings is shown for low CSNR if the channel is Gaussian regardless of the source and, at the other extreme, for high CSNR if the source is Gaussian, regardless of the channel. The extension to the vector settings is also considered where besides the conditions inherited from the scalar case, additional constraints must be satisfied to ensure linearity of the optimal mappings. The problem of error-control in random linear network coding is considered. A ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V capU is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided. Consider a scenario of broadcasting a common content to a group of cooperating wireless nodes that are within proximity of each other. Nodes in this group may receive partial content from the source due to packet losses over wireless broadcast links. We further consider that packet losses are different for different nodes. The remaining missing content at each node can then be recovered, thanks to cooperation among the nodes. In this context, the minimum amount of time that can guarantee a complete acquisition of the common content at every node is referred to as the completion time. It has been shown that instantly decodable network coding (IDNC) reduces the completion time as compared to no network coding in this scenario. Yet, for applications such as video streaming, not all packets have the same importance and not all users are interested in the same quality of content. This problem is even more interesting when additional, but realistic constraints, such as strict deadline, bandwidth, or limited energy are added to the problem formulation. We assert that direct application of IDNC in such a scenario yields poor performance in terms of content quality and completion time. In this paper, we propose a novel Content-Aware IDNC scheme that improves content quality and network coding opportunities jointly by taking into account significance of each packet towards the desired quality of service (QoS). Our proposed Content-Aware IDNC maximizes the quality under the completion time constraint, and minimizes the completion time under the quality constraint.We demonstrate the benefits of Content-Aware IDNC through simulations.