Some Repeated-Root Constacyclic Codes Over Galois Rings

Abstract

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over $\mathop {\mathrm {GR}}\nolimits (2^{a},m)$ of length $2^{s}$ have been characterized: the ring ${\mathcal{ R}}_{2}(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle$ is a chain ring. Furthermore, these results have been generalized to $\lambda$ -constacyclic codes for any unit $\lambda$ of the form $4z-1$ , $z\in \mathop {\mathrm {GR}}\nolimits (2^{a}, m)$ . In this paper, we study more general cases and investigate all cases, where ${\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle$ is a chain ring. In particular, the necessary and sufficient conditions for the ring ${\mathcal{ R}}_{p}(a,m,\gamma)$ to be a chain ring are obtained. In addition, by using this structure we investigate all $\gamma$ -constacyclic codes over $\mathop {\mathrm {GR}}\nolimits (p^{a},m)$ when ${\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual $\gamma$ -constacyclic codes are also provided. Among others, for any prime $p$ , the structure of ${\mathcal{ R}}_{p}(a,m,\gamma) ={\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle$ is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes.