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.

翻译：给定有限字母表$A$、$A^*$上的拟度量$d$以及非负整数$k$，我们引入关系$\tau_{d,k}\subseteq A^*\times A^*$，使得当$d(x,y)\le k$时$(x,y)\in\tau_{d,k}$成立。变长码的检错能力通过关于$\tau_{d,k}$的条件来表述。针对前缀度量、因子度量以及与任意自由幺半群（反）自同构相关联的拟度量，我们证明可以判定给定的正则变长码是否满足任何此类检错约束。