Compact encoding of stationary Markov sources

We treat the problem of encoding the outputs from a controllable rate source whose output statistics can be accurately described as a stationary finite-state Markov process. The class of encoding schemes considered is restricted to those in which each output from the source is encoded as a separate, uniquely decodable codeword. We present a uniquely decodable scheme in this class which minimizes the ensemble average length of the transmitted message corresponding to any number of outputs from the source, i.e., a compact encoding scheme. The code used will vary with time as a function of the entire past history of the source. The future encoding of the outputs from a sample source with any given past history will be compact also. Although the efficiency of the encoding scheme can be improved by enlarging the class of schemes considered, e.g., by applying the technique presented herein to higher order extensions of the source, this topic is not pursued.