Given a finite alphabet $A$ and a binary relation $τ\subseteq A^*\times A^*$, a set $X$ is $τ$-{\it independent} if $ τ(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 $τ_{d,k}$ defined by $(x,y)\inτ_{d,k}$ 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 $τ_{d,k}$. 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.