Some Repeated-Root Constacyclic Codes over Galois Rings
(1701.00247)Abstract
Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over $GR(2a,m)$ of length $2s$ have been characterized: the ring $\mathcal{R}2(a,m,-1)= \frac{GR(2a,m)[x]}{\langle x{2s}+1\rangle}$ is a chain ring. Furthermore, these results have been generalized to $\lambda$-constacyclic codes for any unit $\lambda$ of the form $4z-1$, $z\in GR(2a, m)$. In this paper, we study more general cases and investigate all cases where $\mathcal{R}p(a,m,\gamma)= \frac{GR(pa,m)[x]}{\langle x{ps}-\gamma \rangle}$ 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 $GR(pa,m)$ 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 $\mathcal{R}_p(a,m,\gamma)=\frac{GR(pa,m)[x]}{\langle x{ps}-\gamma\rangle}$ is used to establish the Hamming and homogeneous distances of $\gamma$-constacyclic codes.
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