  Given a finite alphabet $A$ and a binary relation $\tau\subseteq A^*\times
A^*$, a set $X$ is $\tau$-{\it independent} if $ \tau(X)\cap X=\emptyset$.
Given a quasi-metric $d$ over $A^*$ (in the meaning of \cite{W31}) and $k\ge
1$, we associate the relation $\tau_{d,k}$ defined by $(x,y)\in\tau_{d,k}$ if,
and only if, $d(x,y)\le k$ \cite{CP02}.In the spirit of \cite{JK97,N21}, the
error detection-correction capability of variable-length codes can be expressed
in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the
factor one, and every quasi-metric associated to (anti-)automorphisms of the
free monoid, we examine whether those conditions are decidable for a given
regular code.
