Let a and m be positive integers and \(\lambda\) be any unit in \({{\,\mathrm{GR}\,}}(p^a,m)\) of the form \(\lambda =(\sigma _0+p\sigma _1+p^2z)\), where \(\sigma _0,\sigma _1 \in \mathcal {T}(p,m)\smallsetminus \{0\}\) and \(z \in {{\,\mathrm{GR}\,}}(p^a,m)\). The symbol-pair distance of all such \(\lambda\)-constacyclic codes over \({{\,\mathrm{GR}\,}}(p^a,m)\) of length \(p^s\) are determined. As an application, we identify all maximum distance separable (MDS) \(\lambda\)-constacyclic codes of length \(p^s\) over \({{\,\mathrm{GR}\,}}(p^a,m)\) with respect to the symbol-pair distance. We give numerous examples of newly constructed MDS symbol-pair codes, i.e., new optimal symbol-pair codes with respect to the Singleton bound.

设a和m为正整数，\(\lambda\)为\({{\,\mathrm{GR}\,}}(p^a,m)\)中的任意单位，形式为\(\lambda = (\sigma _0+p\sigma _1+p^2z)\)，其中\(\sigma _0,\sigma _1 \in \mathcal {T}(p,m)\smallsetminus \{0\}\)和\ (z \in {{\,\mathrm{GR}\,}}(p^a,m)\)。所有此类\(\lambda\) - 常数循环码在长度为\(p^s\ )上的符号对距离)被确定。作为一个应用程序，我们识别所有最大距离可分离 (MDS) \(\lambda\) - 长度为\(p^s\ )的常数循环码\({{\,\mathrm{GR}\,}}(p^a,m)\)相对于符号对距离。我们给出了许多新构建的 MDS 符号对代码的例子，即关于单例界的新的最优符号对代码。