On Symbol-Triple Distance of a Class of Constacyclic Codes of Length 3ps Over Fpm + uFpm

Let $p\not =3$ be any prime. In this paper, we completely determine symbol-triple distance of all $\gamma $ -constacyclic codes of length $3p^{s}$ over the finite commutative chain ring ${\mathcal{ R}}=\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ , where $\gamma $ is a unit of $\cal R$ which is not a cube in $\mathbb {F}_{p^{m}}$ . We give the necessary and sufficient condition for a symbol-triple $\gamma $ -constacyclic code to be an MDS symbol-triple code. Using that, we establish all MDS symbol-triple $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ . Some examples of the symbol-triple distance of $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ are provided. We also list some new MDS symbol-triple $\gamma $ -constacyclic codes of length $3p^{s}$ over $\cal R$ , where $\gamma $ is not a cube in $\mathbb {F}_{p^{m}}$ .