Journal of Applied Mathematics and Computing

Construction of quantum codes from $$\lambda $$-constacyclic codes over the ring $$\frac{\mathbb {F}_p[u,v]}{\langle v^3-v , u^3-u , uv-vu\rangle }$$

Karthick Gowdhaman, Cruz Mohan, Chinnapillai Durairajan, Jian Gao

Published Aug 9, 2020• 10 citations

DOI: 10.1007/s12190-020-01407-7

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Journal of Applied Mathematics and Computing

Construction of quantum codes from $\lambda$ -constacyclic codes over the ring $\frac{\mathbb {F}_p[u,v]}{\langle v^3-v , u^3-u , uv-vu\rangle }$ Construction of quantum codes from $$\lambda $$-constacyclic codes over the ring $$\frac{\mathbb {F}_p[u,v]}{\langle v^3-v , u^3-u , uv-vu\rangle }$$

Karthick Gowdhaman, Cruz Mohan, Chinnapillai Durairajan, Jian Gao

Published Aug 9, 2020• 10 citations

Publisher

DOI: 10.1007/s12190-020-01407-7

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The goal of this work is the construction of quantum codes over $\mathbb{F}_{q}$ by using $\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+$ $\mu u v^{2}+\nu u^{2} v^{2}$ -constacyclic codes over the ring $R=\mathbb{F}_{q}+\mathbb{F}_{q} u+$ $\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}$ . We give the structure of $\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+\mu u v^{2}+\nu u^{2} v^{2}$ . constacyclic and obtain self-orthogonal codes. We decompose a constacyclic code over $\mathbb{F}_{q}+\mathbb{F}_{q} u+\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+$ $\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}$ into constacyclic codes over $\mathbb{F}_{q}$ . This decomposition makes it possible to give the parameters of the corresponding quantum code.The goal of this work is the construction of quantum codes over $\mathbb{F}_{q}$ by using $\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+$ $\mu u v^{2}+\nu u^{2} v^{2}$-constacyclic codes over the ring $R=\mathbb{F}_{q}+\mathbb{F}_{q} u+$ $\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}$. We give the structure of $\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+\mu u v^{2}+\nu u^{2} v^{2}$. constacyclic and obtain self-orthogonal codes. We decompose a constacyclic code over $\mathbb{F}_{q}+\mathbb{F}_{q} u+\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+$ $\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}$ into constacyclic codes over $\mathbb{F}_{q}$. This decomposition makes it possible to give the parameters of the corresponding quantum code.

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Construction of quantum codes from

\lambda

-constacyclic codes over the ring

\frac{\mathbb {F}_p[u,v]}{\langle v^3-v , u^3-u , uv-vu\rangle }

Constacyclic codes over the ring

{\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q

F q + v F q + v 2 F q and their applications of constructing new non-binary quantum codes

Quantum Codes Obtained from Some Constacyclic Codes Over a Family of Finite Rings

F_{p}+uF_{p}+vF_{p}

Constacyclic codes over

F_q[u_{1}, u_{2}, \cdots , u_{k}]/\langle u_{i}^3-u_{i},u_{i}u_{j}-u_{j}u_{i}\rangle

and their applications of constructing quantum codes

The goal of this work is the construction of quantum codes over

\mathbb{F}_{q}

by using

\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+

\mu u v^{2}+\nu u^{2} v^{2}

-constacyclic codes over the ring

R=\mathbb{F}_{q}+\mathbb{F}_{q} u+

\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}

. We give the structure of

\alpha+\beta u+\gamma v+\delta u v+\eta u^{2}+\theta v^{2}+\lambda u^{2} v+\mu u v^{2}+\nu u^{2} v^{2}

. constacyclic and obtain self-orthogonal codes. We decompose a constacyclic code over

\mathbb{F}_{q}+\mathbb{F}_{q} u+\mathbb{F}_{q} v+\mathbb{F}_{q} u v+\mathbb{F}_{q} u^{2}+\mathbb{F}_{q} v^{2}+

\mathbb{F}_{q} u^{2} v+\mathbb{F}_{q} u v^{2}+\mathbb{F}_{q} u^{2} v^{2}

into constacyclic codes over

\mathbb{F}_{q}

. This decomposition makes it possible to give the parameters of the corresponding quantum code.

Constacyclic codes over

{\pmb {\mathbb {F}}}_{q^2}[u]/\langle u^2-w^2 \rangle

and their application in quantum code construction