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A new upper bound Ru(D) and lower bound Rl(D) are developed for the rate-distortion function of a binary symmetric Markov source with respect to the frequency of error criterion. Both bounds are explicit in the sense that they do not depend on a blocklength parameter. In the interval Dc less than D less than 1/2=Dmax, where Dc is Gray's critical value of distortion, Ru(D) is convex downward and possesses the correct value and the correct slope at both endpoints. The new lower bound Rl(D) diverges from the Shannon lower bound at the same value of distortion as does the second-order Wyner-Ziv lower bound. However, it remains strictly positive for all D equal to or less than 1/2 and therefore eventually rises above all the Wyner-Ziv lower bounds as D approaches 1/2. Some generalisations suggested by the analytical and geometrical techniques employed to derive Ru(D) and Rl(D) are discussed.