Explicit bounds to R(D) for a binary symmetric Markov source

A new upper hound
R_{u}(D)
and lower hound
R_{\ell}(D)
are developed for the rate-distortion function of a binary symmetric Markov source with respect to the frequency of error criterion. Both hounds are explicit in the sense that they do not depend on a blocklength parameter. In the interval
D_{c} < D < 1/2 = D_{max}
, where
D_{c}
is Gray's critical value of distortion,
R_{u}(D)
is convex downward and possesses the correct value and the correct slope at both endpoints. The new lower bound
R_{ell}(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 \leq 1/2
and therefore eventually rises above all the Wyner-Ziv lower bounds as
D
approaches
1/2
. Some generalizations suggested by the analytical and geometrical techniques employed to derive
R_{u}(D)
and
R_{ell}(D)
are discussed.