Lecture notes in networks and systems

Quantum Codes from Cyclic Codes over the Ring $$R = \mathbbm {F}_{q} + u\mathbbm {F}_{q} + v\mathbbm {F}_{q} +uv \mathbbm {F}_{q} + v^{2}\mathbbm {F}_{q}+ uv^{2}\mathbbm {F}_{q}$$

Mohammed Sabiri, Bassou Aouijil

Published Jan 1, 2024• 0 citations

DOI: 10.1007/978-981-97-3305-7_3

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On Quantum and LCD Codes from the Consatacyclic and Cyclic Codes Over the Ring $$\mathbb {F}_{p^{r}}[v]/< v^{m} - v>$$

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Lecture notes in networks and systems

Quantum Codes from Cyclic Codes over the Ring $$R = \mathbbm {F}_{q} + u\mathbbm {F}_{q} + v\mathbbm {F}_{q} +uv \mathbbm {F}_{q} + v^{2}\mathbbm {F}_{q}+ uv^{2}\mathbbm {F}_{q}$$

Mohammed Sabiri, Bassou Aouijil

Published Jan 1, 2024• 0 citations

Publisher

DOI: 10.1007/978-981-97-3305-7_3

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On quantum and LCD Codes Over The Ring F q [u, v, w, z] /

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Mohammed Sabiri, Bassou Aouijil

Lecture notes in networks and systemsJan 1, 20250 citations

Quantum codes from cyclic codes over the ring

{\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q

F q + v 1 F q + ⋯ + v r F q

We give a construction of quantum codes over

{\mathbb{F}_q}

from cyclic codes over a finite non-chain ring

{\mathbb{F}_q} + v{\mathbb{F}_q} + {v^2}{\mathbb{F}_q} + {v^3}{\mathbb{F}_q} + {v^4}{\mathbb{F}_q}

, where q=pr, p is an odd prime, 4|p−1 and v5=v.

On quantum and LCD codes from the cyclic codes over the ring

\mathbb {F}_q[\textsf{u}, \textsf{v},\textsf{w}]/\langle \textsf{u}^3-\textsf{u},\textsf{v}^2-\textsf{v}, \textsf{w}^2-\textsf{w}, \textsf{u}\textsf{v}, \textsf{v}\textsf{u}, \textsf{u}\textsf{w}, \textsf{w}\textsf{u}, \textsf{v}\textsf{w}-\textsf{w}\textsf{v} \rangle

For an odd prime power q and

{\mathcal{R}}_{7}=\mathbb{F}_{q}+u \mathbb{F}_{q}+

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

. Through this work we seek to construct new quantum codes and LCD codes by means of cyclic codes built on the ring

{\mathcal{R}}_{7}

. By using cyclic codes containing their duals, we give quantum codes, we also give a method for the construction of LCD Codes over

{\mathcal{R}}_{7}

On Quantum and LCD Codes from the Consatacyclic and Cyclic Codes Over the Ring

\mathbb {F}_{p^{r}}[v]/< v^{m} - v>