International Conference on Modeling, Simulation, and Applied Optimization

Quantum codes from cyclic codes over a class of nonchain rings

Mustafa Sarı, İrfan Şiap

Published May 1, 2015• 3 citations

Abstract

Lately, quantum error correcting codes have been considered as the Gray images of some special codes over various rings. In this study, we investigate the structure of the cyclic codes of arbitrary length over the finite nonchain ring Rp= Fp+vFp+···+vp-1Fpwhere vp= v and obtain quantum error correcting codes over Fpby using the Gray map defined in the work of Bayram and Siap (2014).

DOI: 10.1109/icmsao.2015.7152240

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International Conference on Modeling, Simulation, and Applied Optimization

Quantum codes from cyclic codes over a class of nonchain rings

Mustafa Sarı, İrfan Şiap

Published May 1, 2015• 3 citations

Publisher

Abstract

DOI: 10.1109/icmsao.2015.7152240

Related papers

Construction of quantum codes from cyclic codes over Fp + vFp

Mohammad Ashraf, Ghulam Mohammad

International Journal of Information and Coding TheoryJan 1, 201534 citations

ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS

Mustafa Sarı, İrfan Şiap

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Abdullah Dertli, Yasemin Çengellenmiş, Şenol Eren

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Encoding and Construction of Quantum Codes from $(γ,Δ)$ -cyclic Codes over a Class of Non-chain RingsEncoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings

Om Prakash, Shikha Patel, Habibul Islam

Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$ -cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$ , $Δ$ is a $γ$ -derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$ . Here, we show that a $(γ,Δ)$ -cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$ -cyclic codes of length $n$ over $\mathbb{F}_q$ , where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$ -derivation of $\mathbb{F}_q$ . Further, necessary and sufficient conditions for both $(γ,Δ)$ -cyclic and $(θ,\Im)$ -cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.

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In this paper, we extend the results given in [3] to a nonchain ring

R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p

, where

v^p=v

and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring

R_p

and study the structure of their duals. We classify cyclic codes containing their duals over

R_p

by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map

{\pi}

defined in [4], we give the parameters of the quantum codes of length pn over

{\mathbb{F}}_p

which are obtained from cyclic codes over

R_p

. Finally, we illustrate the results by giving some examples.

Encoding and Construction of Quantum Codes from

(γ,Δ)

-cyclic Codes over a Class of Non-chain Rings

Let

\mathbb{F}_q

be a finite field of

q=p^m

elements where

p

is a prime and

m

is a positive integer. This paper considers

(γ,Δ)

-cyclic codes over a class of finite non-chain commutative rings

\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle

where

γ

is an automorphism of

\mathscr{R}_{q,s}

Δ

is a

γ

-derivation of

\mathscr{R}_{q,s}

and

1\leq i\neq j\leq s

for a positive integer

s

. Here, we show that a

(γ,Δ)

-cyclic code of length

n

over

\mathscr{R}_{q,s}

is the direct sum of

(θ,\Im)

-cyclic codes of length

n

over

\mathbb{F}_q

, where

θ

is an automorphism of

\mathbb{F}_q

and

\Im

is a

θ

-derivation of

\mathbb{F}_q

. Further, necessary and sufficient conditions for both

(γ,Δ)

-cyclic and

(θ,\Im)

-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.