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

Entropy

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

Enbin Zhang, Bo Hyun Kong, Xiying Zheng

Published Apr 3, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.3390/e28040407

Related papers

Constacyclic Codes over Group Ring (Zq[v])/G

Alireza Soleimani

Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G

Theory of Approximation and Applications•Mar 1, 2019•0 citations

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

Mohammed Sabiri

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 $\mathcal{T}$.

International Conference on Advanced Communication Technologies and Networking (CommNet)•Dec 3, 2025•0 citations

Hulls of Non-Separable Constacyclic Codes over FqR and Their Applications in Constructing New QECCs

Xiying Zheng, Bo Hyun Kong, Enbin Zhang

Let R=Fq[u1,u2,u3]/⟨ui2=ui,uiuj=ujui=0⟩ and FqR={(a,d)|a∈Fq,d∈R}, where q=pm, p is a prime, and m∈N+. We investigate the algebraic structure of the non-separable constacyclic codes over the mixed-alphabet ring FqR and then determine the generator polynomials and the dimensions of their hulls. As an application, using the construction X, we obtain some new quantum error-correcting codes that are not listed in the online quantum codes database.

Entropy•Sep 17, 2026•0 citations

Quantum codes over rings

Kenza Guenda, Thomas Aaron Gulliver

This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽pk are also obtained from linear codes over rings using the generalized Gray map.

International Journal of Quantum Information•Jun 1, 2014•17 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.

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
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.
Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was first introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to find many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G
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