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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 th-order Markov sources with , there is a critical rate such that strong universal codes exist for rates greater than or equal to , but not for rates less than .