Constacyclic codes of length $4p^s$ over the Galois ring $GR(p^a,m)$

For prime $p$, $GR(p^a,m)$ represents the Galois ring of order $p^{am}$ and characterise $p$, where $a$ is any positive integer. In this article, we study the Type (1) $λ$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $λ=ξ_0+pξ_1+p^2z$, $ξ_0,ξ_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $λ$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,λ)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-λ\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-δ\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+δ\rangle}$. In second, when $λ$ is not a square, we show that $\mathcal{R}_p(a,m,λ)$ is a chain ring whose ideals are $\langle (x^4-α)^i\rangle\subseteq \mathcal{R}_p(a,m,λ)$, for $0\leq i\leq ap^s$ where $α^{p^s}=ξ_0$. Also, we prove the dual of the above code is $\langle (x^4-α^{-1})^{ap^s-i}\rangle\subseteq \mathcal{R}_p(a,m,λ^{-1})$ and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) $λ$-constacyclic codes of length $4p^s$ are obtained when $λ$ is not a square.

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