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

TL;DR

This paper characterizes Type (1) λ-constacyclic codes of length 4p^s over Galois rings, detailing their structure, duals, self-orthogonality, self-duality, and distance properties, depending on whether λ is a square.

Contribution

It provides a complete structural analysis of these codes over Galois rings, including duality and distance metrics, for the first time.

Findings

Ideal decomposition when λ is a square

Chain ring structure when λ is not a square

Distance and weight distribution results

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 $\lambda=\xi_0+p\xi_1+p^2z$, $\xi_0,\xi_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $\lambda$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,\lambda)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-\lambda\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-\delta\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+\delta\rangle}$. In second, when $\lambda$ is not a square, we show that $\mathcal{R}_p(a,m,\lambda)$ 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.