International Conference on Advanced Communication Technologies and Networking (CommNet)

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

Mohammed Sabiri, Bassou Aouijil

Published Dec 4, 2024• 0 citations

Abstract

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 ${\mathcal{R}}_{7}$.

DOI: 10.1109/commnet63022.2024.10793302

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International Conference on Advanced Communication Technologies and Networking (CommNet)

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

Mohammed Sabiri, Bassou Aouijil

Published Dec 4, 2024• 0 citations

Publisher

Abstract

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}

DOI: 10.1109/commnet63022.2024.10793302

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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 the Structure of Cyclic Codes Over

\mathbb{F}_{q}{RS}

and Applications in Quantum and LCD Codes Constructions

For a prime p and a positive integer e, let

\mathbb {F}_{p^{e}}

be the finite field of order

p^{e}

and$\mathfrak {R}_{\ell }

be a non-chain ring given by

\mathfrak {R}_{\ell } :=\mathbb {F}_{p^{e}}[v]/\langle v^{\ell } -1\rangle,~\ell \gt 1

. This study presents the

(\theta _{1},\delta _{1},\beta _{1})

-cyclic codes over

\mathfrak {R}_{\ell }

and

(\theta _{i},\delta _{i},\beta _{i})

-cyclic codes over

\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }

for

i=0,1

. Further, we study the application of these codes in the construction of quantum codes. Towards this, we define a Gray map to find images of the elements of

\mathfrak {R}_{\ell }

and

\mathbb {F}_{p^{e}}\mathfrak {R}_{\ell }

over the copies of

\mathbb {F}_{p^{e}}$, and then establish the dual-containing conditions for these codes. Finally, using the CSS construction and a propagation rule, we find many non-binary quantum and classical codes with better parameters.

Let

{\mathcal{R}} = {{\mathbb{F}}_q}[u,v]/,

where

{{\mathbb{F}}_q}

is a finite of cardinality q and u2= u,v3= v,uv = vu. In this work, we investigate quantum and LCD codes over

{{\mathbb{F}}_q}

by using constacyclic codes over the ring

{\mathcal{R}}

, for the first codes, dual or not dual containing codes were used, and for the second codes, constacylic LCD codes were used in order to produce good codes over finite fields.

Let

\mathcal{T} = {\mathbb{F}_{{p^s}}} + {u_1}{\mathbb{F}_{{p^s}}} + {u_2}{\mathbb{F}_{{p^s}}} + {u_3}{\mathbb{F}_{{p^s}}} + u_1^2{\mathbb{F}_{{p^s}}}

be a finite non-chain ring. In this paper, we propose a method of constructing quantum codes and LCD codes from constacyclic codes over

\mathcal{T}

. The first codes are obtained via the Gray map and the Calderbank-Shor-Steane construction from Euclidean dualcontaining constacyclic codes over

\mathcal{T}

. The second codes are obtained as the gray image of consta cyclic codes of LCD codes over

\mathcal{T}