On Constacyclic Codes of Length $4p^s$ over $F_{p^m}[u,v]/\langle u^2, v^2, uv-vu \rangle$

Abstract

In this paper, we determine the algebraic structure of all $\lambda$-constacyclic codes of length $4p^s$ over the ring $R_{u^2,v^2,p^m} = F_{p^m}[u,v]/\langle u^2, v^2, uv-vu \rangle$ and $u^2 = 0, v^2 = 0$ where $\lambda = (\alpha+\beta u+\gamma v+\delta uv)$ with $\beta,\gamma,\delta \in F_{p^m}$, $\alpha \in F_{p^m}^*$ and $\beta, \gamma$ are not both zero. If $\lambda$ is a square, each $\lambda$-constacyclic codes of length $4p^s$ is expressed as a direct sum of an-$\alpha$-constacyclic code and $\alpha$-constacyclic code of length $2p^s$. In the primary case where the unit $\lambda$ is not square, it is shown that any non-zero polynomial of degree $\leq 4$ over $F_{p^m}$ is invertible in the ring $R_{\alpha,\beta,\gamma,\delta} = R_{u^2,v^2,p^m}[x]/\langle x^{4p^s}-\lambda \rangle$ when $\lambda = (\alpha+\beta u+\gamma v +\delta uv)$ for non-zero elements $\alpha, \beta,\gamma \in F_{p^m}^*$, $\delta \in F_{p^m}$, it follows that the ring $R_{\alpha,\beta,\gamma,\delta}$ is a chain ring with maximal ideal $(x^4-\alpha_0)$ and $(\alpha+\beta u+\gamma v+\delta uv)$-constacyclic codes are $\langle (x^4-\alpha_0)^i \rangle$, for $0 \leq i \leq 2p^s$. In the second case where $\lambda$ is not square and $\lambda =\gamma$ for $\gamma \in F_{p^m}$, it is obtain that $R_{u^2,v^2,p^m}[x]/\langle x^{4p^s}-\gamma \rangle$ is a local ring with maximal ideal $\langle x^4-\gamma_0,u \rangle$ but not a chain ring, such $\lambda$-constacyclic codes are classified into four distinct types. We provide the number of codewords for each type, and give the details of their dual codes.