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| On Constacyclic Codes of Length $4p^{s}$ over $F_{p^{m}}[u,v]/\langle u^2,v^2,uv-vu \rangle$ |
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| KeyWord:Constacyclic codes, codes over rings, dual codes |
| Author Name | Affiliation | | Somaiyah A. A. Abdulsattar | School of Mathematics Sciences, SRTM University, Vishnupuri, Nanded, 431606, Maharashtra, India | | Arunkumar Patil | Department of Mathematics, SGGS Institute of Engineering and Technology, Vishnupuri, Nanded, 431606, Maharashtra, India |
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| 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. |
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