A **coded system** [see F. Blanchard, G. Hansel, *Systèmes codés*, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8.
(http://www.sciencedirect.com/science/article/pii/0304397586901088)] is a shift space $X$ which admits a *cover* (a *presentation*) by a countable labeled strongly connected (irreducible) directed graph $(G, g)$ where a labeling function maps the edge set of $G$ into a finite set $A$. This means that $X$ is the closure of the image of the map $L_g \colon S_G \to A^\mathbb{Z}$ induced on the countable Markov shift
$S_G$ of all bi-infinite paths in G by $L_g(x)_i := g(x_i)$,
$i \in \mathbb{Z}$, $x \in S_G$. In Lind and Marcus book *An introduction to symbolic dynamics and coding*, page 451 there are some other equivalent conditions.

A coded system is transitive by definition. A **synchronized system** is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$ then $uvw$ is also admissible. Every transitive sofic shift is synchronized. A synchronized system is well-known to be coded. Moreover, any synchronized system admits a presentetion by a connected, right-resolving, follower separated labeled graph with a magic word (known as *the* Fisher cover).

It is relatively easy to prove that a synchronized system is topologically mixing if and only if there are two closed paths in its Fisher cover of relatively prime lengths. Does the same hold for coded systems? It is easy to see that two closed paths in $G$ with relatively prime lengths imply that the shift space presented by $G$ is mixing, but how about the converse? I mean, does every mixing coded system has a presentation with two closed paths of relatively prime lengths?

**Note:** The equivalence holds if $G$ is finite, but I am interested in the case, when $G$ may be infinite.

**Edit** There exists a presentation of a full shift such that every loop has even length. But there are presentations of the full shift with two closed paths of relatively prime lengths.