Finite Blocklength Entropy-Achieving Coding for Linear System Stabilization

In this article, we consider the minimum data rate problem for linear system stabilization under noiseless communication channels. Previous results indicated that having a data rate very approaching the entropy bound leads to large delays and data buffer sizes. In analogy, the entropy bound in Shannon's source coding theorem in traditional information theory displays this behavior, where the data rate can be arbitrarily close to the entropy bound but only at the cost of boundlessly enlarging the blocklength. However, in this article, we show the analogy is not strict. We prove that it is possible to stabilize a linear system at a rate equal to the entropy bound with zero delay, i.e., where each system state is encoded and decoded within one time unit. We establish a set of sufficient conditions for guaranteeing zero-delay entropy-achieving codes. Following this, we design an entropy-achieving code with finite blocklength satisfying the set of sufficient conditions, where the codeword length is uniformly bounded.