  Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a
non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq
A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The
error detection capability of variable-length codes is expressed in term of
conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor
one, and any quasi-metric associated with some free monoid (anti-)automorphism,
we prove that one can decide whether a given regular variable-length code
satisfies any of those error detection constraints.
