Functional Index Coding, Network Function Computation, and Sum-Product Algorithm for Decoding Network Codes

Abstract

Network coding was introduced as a means to increase throughput in communication networks when compared to routing. Network coding can be used not only to communicate messages from some nodes in the network to other nodes but are also useful when some nodes in a network are interested in computing some functions of information generated at some other nodes. Such a situation arises in sensor networks. In this work, we study three problems in network coding. First, we consider the functional source coding with side information problem wherein there is one source that generates a set of messages and one receiver which knows some functions of source messages and demands some other functions of source messages. Cognizant of the receiver's side information, the source aims to satisfy the demands of the receiver by making minimum number of coded transmissions over a noiseless channel. We use row-Latin rectangles to obtain optimal codes for a given functional source coding with side information problem. Next, we consider the multiple receiver extension of this problem, called the functional index coding problem, in which there are multiple receivers, each knowing and demanding disjoint sets of functions of source messages. The source broadcasts coded messages, called a functional index code, over a noiseless channel. For a given functional index coding problem, the restrictions the demands of the receivers pose on the code are represented using the generalized exclusive laws and it is shown that a code can be obtained using the confusion graph constructed using these laws. We present bounds on the size of an optimal code based on the parameters of the confusion graph. For the case of noisy broadcast channel, we provide a necessary and sufficient condition that a code must satisfy for correct decoding of desired functions at each receiver and obtain a lower bound on the length of an error-correcting functional index code. In the second problem, we explore relation between network function computation problems and functional index coding and Metroid representation problems. In a network computation problem, the demands of the sink nodes in a directed acyclic multichip network include functions of the source messages. We show that any network computation problem can be converted into a functional index coding problem and vice versa. We prove that a network code that satisfies all the sink demands in a network computation problem exists if and only if its corresponding functional index coding problem admits a functional index code of a specific length. Next, we establish a relation between network computation problems and representable mastoids. We show that a network computation problem in which the sinks demand linear functions of source messages admits a scalar linear solution if and only if it is matricidal with respect to a representable Metroid whose representation fulfils certain constraints dictated by the network computation problem. Finally, we study the usage of the sum-product (SP) algorithm for decoding network codes. Though lot of methods to obtain network codes exist, the decoding procedure and complexity have not received much attention. We propose a SP algorithm based decoder for network codes which can be used to decode both linear and nonlinear network codes. We pose the decoding problem at a sink node as a marginalize a product function (MPF) problem over the Boolean smearing and use the SP algorithm on a suitably constructed factor graph to perform decoding. We propose and demonstrate the usage of trace back to reduce the number of operations required to perform SP decoding. The computational complexity of performing SP decoding with and without trace back is obtained. For nonlinear network codes, we define fast decidability of a network code at sinks that demand all the source messages and identify a sufficient condition for the same. Next, for network function computation problems, we present an MPF formulation for function computation at a sink node and use the SP algorithm to obtain the value of the demanded function.