On the Decodable Probability Bound of Linear Network Coding in Acyclic Lossy Networks

This paper presents new analytical results of linear network coding in acyclic lossy networks. Network coding in lossy networks can be characterized by three kernels: 1.) local encoding kernel; 2.) global encoding kernel; and 3.) successful transmission probability (STP) kernel. A STP kernel of each channel provides the accumulated successful transmission probability of transmitted data from a source to the considering channel. At a specific intermediate node, STP kernels corresponding to outgoing channels are computed from STP kernels corresponding to incoming channels and successful transmission probabilities of outgoing channels. Based on the random matrix theory, the probability bound on the random selection of global encoding kernels allowing linear network coded data can be decoded at destination is derived. The derived bound is a function of a field size and a dimension of global encoding kernel. Linear network coded data arriving at destinations can be viewed as random variables and form a random matrix characterized by STP kernels. With the random matrix of arriving data, the probability bound in decoding all transmitted data perfectly is computed.