Capacity of Higher-Dimensional Constrained Systems

One-dimensional constrained systems, also known as discrete noiseless channels and sofic shifts, have a well-developed theory and have played an important role in applications such as modulation coding for data recording. Shannon found a closed form expression for the capacity of such systems in his seminal paper, and capacity has served as a benchmark for the efficiency of coding schemes as well as a guide for code construction. Advanced data recording technologies, such as holographic recording, may require higher-dimensional constrained coding. However, in higher dimensions, there is no known general closed form expression for capacity. In fact, the exact capacity is known for only a few higher-dimensional constrained systems. Nevertheless, there have been many good methods for efficiently approximating capacity for some classes of constrained systems. These include transfer matrix and spatial mixing methods. In this article, we will survey progress on these and other methods.