During a course about finite dynamical systems the following conjecture was presented to us :

Let G be a directed graph of order n.
Let $\tau(G)$ be the minimum size of a subset of $V(G)$, $I$ such that $G \setminus I$ is acyclic (Feedback Vertex Set). Let $g(G)$ be the girth of $G$ (the minimum size of a cycle of $G$). If G is acyclic, let $g(G) = n+1$.
Let $A(n,d)$ be the maximum size of a subset $X \subseteq \{0,1\}^n$ such that the Hamming distance of two distinct elements of $X$ is always larger or equal than $d$.

Conjecture : for all graphs $G$ of order $n$, $2^{\tau(G)}\leqslant A(n,g(G))$.

From my understanding, this is just an attempt to find properties of codes and of some particular graphs describing $A(n,d)$ (e.g. hypercubes of dimention $n$ for $A(n,1)$ or subgraphs of these hypercubes for $A(n,x)$). Is that true ?

Does $A(n,g(G))$ describe something useful in particular (in this case $G$ seems to describe the length of the words while it described the words themselves in the case of hypercubes) ?

Are there other consequences/implications ? Does the fact that G should be directed matter ?

Thank you in advance.