On the Decodability of Random Linear Network Coding in Acyclic Networks

This paper presents novel analytical results on the successful decoding probability for random linear network coding in acyclic networks. The results consist of a tight lower bound on the successful decoding probability, its convergence, and its application in constructing a practical algorithm to identify the minimum field size for random linear network coding subject to a target on the successful decoding probability. From the two characterizations of random linear network coding, namely the set of local encoding kernels and the set of global encoding kernels, we first show that choosing randomly and uniformly the coefficients of the local encoding kernels results in uniform and independent coefficients for the global encoding kernels. The set of global encoding kernels for an arbitrary destination is thus a random matrix whose invertibility is equivalent to decodability. The lower bound on the successful decoding probability is then derived in terms of the probability that this random matrix is non-singular. The derived bound is a function of the field size and the dimension of global encoding kernels. The convergence rates of the bound over these two parameters are provided. Compared to the mathematical expression of the exact probability, the derived bound provides a more compact expression and is close to the exact value. As a benefit of the bound, we construct a practical algorithm to identify the minimum field size in order to achieve a target on the successful decoding probability. Simulation and numerical results verify the validity of the derived bound as well as its higher precision than previously established bounds.