Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Conference on Advanced Communication Technologies and Networking (CommNet)

Some new Quantum Codes and LCD Codes over a finite non-chain ring

Mohammed Sabiri

Published Dec 3, 2025• 0 citations
Publisher

Abstract

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 $\mathcal{T}$.

DOI
10.1109/commnet68224.2025.11288868

Related papers

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.

Advances in Mathematics of Communications•Jan 1, 2019•46 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 94B99

Quantum Information Processing•Jul 18, 2024•6 citations

Quantum Codes from Galois Hulls of Constacyclic Codes over a Finite Non-Chain Ring

Enbin Zhang, Bo Hyun Kong, Xiying Zheng

We study the algebraic structures of constacyclic codes over a new finite non-chain ring R and their l-Galois dual codes based on a new Gray map ϕ and determine the dimensions of the l-Galois hull of constacyclic codes over R. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over R, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones.

Entropy•Apr 3, 2026•1 citation

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 $\mathbb{F}_{p^m}$.

arXiv:2010.07175•Oct 14, 2020•0 citations

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and application to quantum codes

Yongsheng Tang, Ting Yao, Shixin Zhu +1 more

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

•Dec 6, 2017•0 citations

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and its application to quantum codes

Yongsheng Tang, Ting Yao, Shixin Zhu +1 more

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

arXiv:1712.02081•Dec 6, 2017•0 citations
Let T=Fps+u1Fps+u2Fps+u3Fps+u12Fps\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}}}T=Fps​+u1​Fps​+u2​Fps​+u3​Fps​+u12​Fps​ be a finite non-chain ring. In this paper, we propose a method of constructing quantum codes and LCD codes from constacyclic codes over T\mathcal{T}T. The first codes are obtained via the Gray map and the Calderbank-Shor-Steane construction from Euclidean dualcontaining constacyclic codes over T\mathcal{T}T. The second codes are obtained as the gray image of consta cyclic codes of LCD codes over T\mathcal{T}T.
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 .Inthispaper,weconstructnewnon−binaryquantumcodesfrom(. In this paper, we construct new non-binary quantum codes from (.Inthispaper,weconstructnewnon−binaryquantumcodesfrom( \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R .Wegivethestructureof(. We give the structure of (.Wegivethestructureof( \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 T=Fq[u1,u2,…,ur]/⟨fi(ui),uiuj−ujui⟩i,j=1r\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}^rT=Fq​[u1​,u2​,…,ur​]/⟨fi​(ui​),ui​uj​−uj​ui​⟩i,j=1r​, where q=pmq=p^mq=pm, ppp is some odd prime, mmm is a positive integer and fi(ui)f_i(u_i)fi​(ui​) are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T\mathcal{T}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 T\mathcal{T}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 Fq\mathbb{F}_qFq​. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99
We study the algebraic structures of constacyclic codes over a new finite non-chain ring R and their l-Galois dual codes based on a new Gray map ϕ and determine the dimensions of the l-Galois hull of constacyclic codes over R. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over R, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones.
New non-binary quantum codes from skew constacyclic codes over the ring Fpm+vFpm+v2Fpm\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}Fpm​+vFpm​+v2Fpm​
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥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 R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}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 Fpm\mathbb{F}_{p^m}Fpm​.
A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and application to quantum codes
We introduce a Gray map from F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ to F22m\mathbb{F}_{2}^{2m}F22m​ and study (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m,\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},F2m​+uF2m​, where u2=0.u^{2}=0.u2=0. It is proved that the image of a (1+u)(1+u)(1+u)-constacyclic code length nnn over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ under the Gray map is a distance-invariant quasi-cyclic code of index mmm and length 2mn2mn2mn over F2.\mathbb{F}_{2}.F2​. We also prove that every code of length 2mn2mn2mn which is the Gray image of cyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ of length nnn is permutation equivalent to a binary quasi-cyclic code of index m.m.m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m.\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.F2m​+uF2m​.
A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and its application to quantum codes
We introduce a Gray map from F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ to F22m\mathbb{F}_{2}^{2m}F22m​ and study (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m,\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},F2m​+uF2m​, where u2=0.u^{2}=0.u2=0. It is proved that the image of a (1+u)(1+u)(1+u)-constacyclic code length nnn over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ under the Gray map is a distance-invariant quasi-cyclic code of index mmm and length 2mn2mn2mn over F2.\mathbb{F}_{2}.F2​. We also prove that every code of length 2mn2mn2mn which is the Gray image of cyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ of length nnn is permutation equivalent to a binary quasi-cyclic code of index m.m.m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m.\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.F2m​+uF2m​.