A necessary condition for reliable communication with feedback based upon Lyapunov exponents of dynamical systems

In this paper we use control theoretic techniques to provide a new perspective for understanding what makes a channel coding scheme good. We show that for a broad class of causal feedback channels, including some with memory, the Lyapunov exponent of the transmission function - when it exists - is fundamentally linked to the maximum rate which the scheme can attain. More generally, we can upper-bound the set of achievable rates for a given encoding scheme by considering a generalization of the Lyapunov exponent of the transmission function. By approaching the problem from a dynamical systems perspective, we hope to provide the intuition to motivate the evaluation and design of new coding schemes.