New non-binary quantum codes from skew constacyclic codes over the ring F p m +v{F p m +v 2 F p m .

Ram Krishna Verma, Om Prakash, Ashutosh Kumar Singh

Published Jan 1, 2020• 0 citations

Abstract

Related papers

New non-binary quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}$

Ram Krishna Verma, Om Prakash, Ashutosh Singh

arXiv:2010.07175Oct 14, 20200 citations

Quantum codes from skew constacyclic codes over Fp m + vFp m + v2Fp m

Ram Krishna Verma, Om Prakash, Ashutosh Singh

International Workshop Algebraic and Combinatorial Coding TheoryOct 11, 20204 citations

New quantum codes from skew constacyclic codes

Ram Krishna Verma, Om Prakash, Ashutosh Singh +1 more

Advances in Mathematics of CommunicationsAug 19, 202119 citations

New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

Fanghui Ma, Jian Gao, Fang‐Wei Fu

Advances in Mathematics of CommunicationsJan 1, 201946 citations

Quantum and LCD codes from skew constacyclic codes over a general class of non-chain rings

Pradeep Rai, Bhupendra Singh, Ashok Ji Gupta

Quantum Information ProcessingJul 18, 20246 citations

Quantum codes from constacyclic codes over $S_{k}$

Bo Hyun Kong, Xiying Zheng

EPJ Quantum TechnologyFeb 6, 20237 citations

New non-binary quantum codes from skew constacyclic codes over the ring F p m +v{F p m +v 2 F p m .

Ram Krishna Verma, Om Prakash, Ashutosh Kumar Singh

Published Jan 1, 2020• 0 citations

OpenAlex

Abstract

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring

\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle

where

p

is an odd prime and

m \geq 1

. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring

\mathcal{R}

. Then a necessary and sufficient condition for skew constacyclic codes over

\mathcal{R}

to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over

\mathbb{F}_{p^m}

Related papers

New non-binary quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}$ New non-binary quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}$

Ram Krishna Verma, Om Prakash, Ashutosh Singh

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring $\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle$ where $p$ is an odd prime and $m \geq 1$ . In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring $\mathcal{R}$ . Then a necessary and sufficient condition for skew constacyclic codes over $\mathcal{R}$ to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over $\mathbb{F}_{p^m}$ .In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring $\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle$ where $p$ is an odd prime and $m \geq 1$. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring $\mathcal{R}$. Then a necessary and sufficient condition for skew constacyclic codes over $\mathcal{R}$ to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over $\mathbb{F}_{p^m}$.

arXiv:2010.07175Oct 14, 20200 citations

Quantum codes from skew constacyclic codes over Fp m + vFp m + v2Fp m

Ram Krishna Verma, Om Prakash, Ashutosh Singh

For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.

International Workshop Algebraic and Combinatorial Coding TheoryOct 11, 20204 citations

New quantum codes from skew constacyclic codes

Ram Krishna Verma, Om Prakash, Ashutosh Singh +1 more

For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

Advances in Mathematics of CommunicationsAug 19, 202119 citations

New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

Fanghui Ma, Jian Gao, Fang‐Wei Fu

Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu $. In this paper, we construct new non-binary quantum codes from ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $. We give the structure of ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu $. In this paper, we construct new non-binary quantum codes from ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $. We give the structure of ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.

Advances in Mathematics of CommunicationsJan 1, 201946 citations

Quantum and LCD codes from skew constacyclic codes over a general class of non-chain rings

Pradeep Rai, Bhupendra Singh, Ashok Ji Gupta

Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings $\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^r$ , where $q=p^m$ , $p$ is some odd prime, $m$ is a positive integer and $f_i(u_i)$ are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over $\mathcal{T}$ and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over $\mathcal{T}$ , and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over $\mathbb{F}_q$ . MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings $\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^r$, where $q=p^m$, $p$ is some odd prime, $m$ is a positive integer and $f_i(u_i)$ are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over $\mathcal{T}$ and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over $\mathcal{T}$, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over $\mathbb{F}_q$. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99

Quantum Information ProcessingJul 18, 20246 citations

Quantum codes from constacyclic codes over $S_{k}$ Quantum codes from constacyclic codes over $S_{k}$

Bo Hyun Kong, Xiying Zheng

Abstract Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle $ S k = F q [ u 1 , u 2 , … , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where $1\leq i,j\leq k$ 1 ≤ i , j ≤ k , $q=p^{m}$ q = p m , p is an odd prime. First, we define two new Gray maps $\phi _{k}$ ϕ k and $\varphi _{k}$ φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over $\mathbb{F}_{q}$ F q by using CSS construction, and compare the constructed codes better than the existing literature.Abstract Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle $ S k = F q [ u 1 , u 2 , … , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where $1\leq i,j\leq k$ 1 ≤ i , j ≤ k , $q=p^{m}$ q = p m , p is an odd prime. First, we define two new Gray maps $\phi _{k}$ ϕ k and $\varphi _{k}$ φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over $\mathbb{F}_{q}$ F q by using CSS construction, and compare the constructed codes better than the existing literature.

EPJ Quantum TechnologyFeb 6, 20237 citations

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring

\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle

where

p

is an odd prime and

m \geq 1

. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring

\mathcal{R}

. Then a necessary and sufficient condition for skew constacyclic codes over

\mathcal{R}

to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over

\mathbb{F}_{p^m}

New non-binary quantum codes from skew constacyclic codes over the ring

\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring

\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle

where

p

is an odd prime and

m \geq 1

. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring

\mathcal{R}

. Then a necessary and sufficient condition for skew constacyclic codes over

\mathcal{R}

to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over

\mathbb{F}_{p^m}

Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu

. In this paper, we construct new non-binary quantum codes from (

\alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R

. We give the structure of (

\alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.

Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings

\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^r

, where

q=p^m

p

is some odd prime,

m

is a positive integer and

f_i(u_i)

are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over

\mathcal{T}

and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over

\mathcal{T}

, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over

\mathbb{F}_q

. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99

Quantum codes from constacyclic codes over

S_{k}

1\leq i,j\leq k

1 ≤ i , j ≤ k ,

q=p^{m}

q = p m , p is an odd prime. First, we define two new Gray maps

\phi _{k}

ϕ k and

\varphi _{k}

φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over

\mathbb{F}_{q}

F q by using CSS construction, and compare the constructed codes better than the existing literature.