Symbol-pair distance of some repeated-root constacyclic codes of length $p^s$ over the Galois ring ${{\,\mathrm{GR}\,}}(p^a,m)$

Abstract

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.