Journal of Physics Conference Series

Quantum Codes From Cyclic Codes Over The Ring R2

Alev Altınel, Murat Güzeltepe

Published Oct 1, 2016• 0 citations

Abstract

Let R 2 denotes the ring F 2 + μF 2 + υ 2 + μυ F 2 + wF 2 + μwF 2 + υwF 2 + μυwF 2 . In this study, we construct quantum codes from cyclic codes over the ring R 2 , for arbitrary length n, with the restrictions μ 2 = 0, υ 2 = 0, w 2 = 0, μυ = υμ, μw = wμ, υw = wυ and μ (υw) = (μυ) w. Also, we give a necessary and sufficient condition for cyclic codes over R 2 that contains its dual. As a final point, we obtain the parameters of quantum error-correcting codes from cyclic codes over R 2 and we give an example of quantum error-correcting codes form cyclic codes over R 2 .

DOI: 10.1088/1742-6596/766/1/012022

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Journal of Physics Conference Series

Quantum Codes From Cyclic Codes Over The Ring R2

Alev Altınel, Murat Güzeltepe

Published Oct 1, 2016• 0 citations

Publisher

Abstract

DOI: 10.1088/1742-6596/766/1/012022

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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

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which are obtained from cyclic codes over

R_p

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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.

QUANTUM CODES FROM CYCLIC CODES OVER

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

In this paper, we investigate the method of constructing quantum codes from cyclic codes over the ring

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

(u2=u, v2=v, uv=vu=0). By means of the generator polynomials and some decomposition of cyclic codes and their dual over R, we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes. Also as an application, we obtain new non binary quantum codes from those classes of cyclic codes.