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

Abstract

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 ${\mathcal{R}}_p(a,m,\lambda )=\frac{GR(p^a,m)[x]}{⟨x^{4p^s}-\lambda ⟩}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{⟨x^{2p^s}-\delta ⟩}$ and $\frac{GR(p^a,m)[x]}{⟨x^{2p^s}+\delta ⟩}$. In second, when $\lambda$ is not a square, we show that ${\mathcal{R}}_p(a,m,\lambda )$ is a chain ring whose ideals are $⟨(x^4-\alpha {)}^i⟩\subseteq {\mathcal{R}}_p(a,m,\lambda )$, for $0\le i\le a{p}^s$ where ${\alpha }^{p^s}={\xi }_0$. Also, we prove the dual of the above code is $⟨(x^4-{\alpha }^{-1}{)}^{a{p}^s-i}⟩\subseteq {\mathcal{R}}_p(a,m,{\lambda }^{-1})$ 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.

There is mistakes in a few initial results that affecting the whole paper