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This paper addresses the problem of bounding the rate-distortion function of a binary symmetric Markov source. We derive a sequence of upper and lower bounds on the rate- distortion function of such sources. The bounds are indexed by k, which corresponds to the dimension of the optimization problem involved. We obtain an explicit bound on the difference between the derived upper and lower bounds as a function of k. This allows to identify the value of k that suffices to compute the rate distortion function to a given desired accuracy. In addition to these bounds, a tighter lower bound which is also a function of k is derived. Our numerical results show that the new bounds improve on the Berger's upper and lower bounds even with small values of k.