I'm reading section 2.8 of the course Principles of Digital Communications I. Suppose I have a Markov chain with a set of state $(a, b, c)$ and $q(a) = q(b) = q(c) = \frac{1}{3}$ (with $q(x)$ is the probability of occurrence $x$) and the state transition is like this:
a -> b: 100%
b -> a: 50%
b -> c: 50%
c -> a: 50%
c -> b: 50%
$H(X|S) = \frac{2}{3}$. Yet, what is the minimum expected code length of this Markov source? $\frac{2}{3}$ or $1$?