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 $GR(2^a,m)$ of length $2^s$ have been characterized: the ring ${\mathcal{R}}_2(a,m,-1)=\frac{GR(2^a,m)[x]}{⟨x^{2^s}+1⟩}$ is a chain ring. Furthermore, these results have been generalized to $\lambda$-constacyclic codes for any unit $\lambda$ of the form $4z-1$, $z\in GR(2^a,m)$. In this paper, we study more general cases and investigate all cases where ${\mathcal{R}}_p(a,m,\gamma )=\frac{GR(p^a,m)[x]}{⟨x^{p^s}-\gamma ⟩}$ is a chain ring. In particular, 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 $GR(p^a,m)$ when ${\mathcal{R}}_p(a,m,\gamma )$ is a chain ring. 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 ${\mathcal{R}}_p(a,m,\gamma )=\frac{GR(p^a,m)[x]}{⟨x^{p^s}-\gamma ⟩}$ is used to establish the Hamming and homogeneous distances of $\gamma$-constacyclic codes.