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) $\lambda$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $\lambda ={\xi }_0+p{\xi }_1+p^{2}z$, ${\xi }_0,{\xi }_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $\lambda$ is a square, we show that any ideal of $$\mathcal{R}$_p(a,m,\lambda)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-\lambda\rangle}$isthedirectsumoftheidealsof$\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-\delta\rangle}$and$\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+\delta\rangle}$\mathrm{.}Insecond,when$\lambda$isnotasquare,weshowthat$$\mathcal{R}$_p(a,m,\lambda)$ is a chain ring whose ideals are $\langle (x^4-\alpha)^i\rangle\subseteq $\mathcal{R}$_p(a,m,\lambda)$,for$0\leq i\leq ap^s$where$\alpha^{p^s}=\xi_0$. Also, we prove the dual of the above code is $\langle (x^4-\alpha^{-1})^{ap^s-i}\rangle\subseteq $\mathcal{R}$_p(a,m,\lambda^{-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) $\lambda$-constacyclic codes of length $4p^s$ are obtained when $\lambda$ is not a square.