Fixed-rate universal codes for Markov sources

The existence of a fixed-rate block source code whose performance for each source in a class of Markov sources is uniformly close to the distortion-rate function of that source is investigated. Such a code is called strong universal. It is found that strong universal codes of all rates exist for the class of all binary first-order Markov sources. But for a larger alphabet or for the class of all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order Markov sources with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n > 2</tex> , there is a critical rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> such that strong universal codes exist for rates greater than or equal to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> , but not for rates less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> .