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

Β© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Informatics and Applied Mathematics

Constacyclic Codes Over 𝑭𝒒[𝒖] /βŒ©π’–πŸ‘ = 𝟎βŒͺ and Their Application of Constructing Quantum Codes

Zineb Hebbache

Published Nov 11, 2022β€’ 0 citations
Publisher

Abstract

Let 𝑅 = πΉπ‘ž+uπΉπ‘ž+𝑒^2πΉπ‘ž, 𝑒^3=0 be a finite chain ring. In this paper, we give the structure of constacyclic codes over 𝑅 and obtain self-orthogonal codes over πΉπ‘ž by using the Gray map from 𝑅𝑛 to πΉπ‘ž^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.

DOI
10.53508/ijiam.1127019

Related papers

Quantum codes from (1 βˆ’ 2v)-constacyclic codes over the ring 𝔽q + u𝔽q + v𝔽q + uv𝔽q

Juan Li, Jian Gao, Yongkang Wang

In this paper, structural properties of [Formula: see text]-constacyclic codes over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained.

Discrete Mathematics Algorithms and Applicationsβ€’Apr 23, 2018β€’23 citations

Quantum codes derived from constacyclic codes over Z3 [u,v]/

Rekha, Monika Sangwan

The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.

International Journal of Statistics and Applied Mathematicsβ€’Aug 1, 2025β€’0 citations

Quantum Codes from Constacyclic Codes over the Ring Fq[u1,u2]/〈 u 1 2 -u1, u 2 2 -u2,u1u2-u2u1βŒͺ

Ahmad N. Al‐Kenani, Mohammad Ashraf, Ghulam Mohammad

In this paper, we study the structural properties of ( Ξ± + u 1 Ξ² + u 2 Ξ³ + u 1 u 2 Ξ΄ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 βˆ’ u 1 , u 2 2 βˆ’ u 2 , u 1 u 2 βˆ’ u 2 u 1 ⟩ where q = p m for odd prime p and m β‰₯ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .

Mathematicsβ€’May 13, 2020β€’7 citations

Construction of some new quantum codes

Mohammed Sabiri

In this paper, we consider$R=\mathbb{F}_{q}[u, v]/ < u^{2}-u, v^{2}-v, uv - vu >$which is a finite non-chain ring, where$q$is an odd prime power and$u^{2}=u, v^{2}=v, uv=vu$. We construct new 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.

International Conference on Advanced Communication Technologies and Networking (CommNet)β€’Dec 12, 2022β€’4 citations

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 Applicationsβ€’Mar 1, 2019β€’0 citations

Quantum Codes Obtained through Constacyclic Codes over Z3 + Ξ½Z3 + Ο‰Z3 + Ξ½Ο‰Z3

Jagbir Singh, Prateek Mor, Shikha . +1 more

This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + Ξ½Z3 + Ο‰Z3 + Ξ½Ο‰Z3 where Ξ½2 = 1, Ο‰2 = Ο‰, Ξ½Ο‰ = Ξ½Ο‰, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.

European Journal of Pure and Applied Mathematicsβ€’Aug 5, 2021β€’2 citations
Let 𝑅 = πΉπ‘ž+uπΉπ‘ž+𝑒^2πΉπ‘ž, 𝑒^3=0 be a finite chain ring. In this paper, we give the structure of constacyclic codes over 𝑅 and obtain self-orthogonal codes over πΉπ‘ž by using the Gray map from 𝑅𝑛 to πΉπ‘ž^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.
The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.
In this paper, we study the structural properties of ( Ξ± + u 1 Ξ² + u 2 Ξ³ + u 1 u 2 Ξ΄ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 βˆ’ u 1 , u 2 2 βˆ’ u 2 , u 1 u 2 βˆ’ u 2 u 1 ⟩ where q = p m for odd prime p and m β‰₯ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .
In this paper, we considerR=Fq[u,v]/<u2βˆ’u,v2βˆ’v,uvβˆ’vu>R=\mathbb{F}_{q}[u, v]/ < u^{2}-u, v^{2}-v, uv - vu >R=Fq​[u,v]/<u2βˆ’u,v2βˆ’v,uvβˆ’vu>which is a finite non-chain ring, whereqqqis an odd prime power andu2=u,v2=v,uv=vuu^{2}=u, v^{2}=v, uv=vuu2=u,v2=v,uv=vu. We construct new quantum codes from(Ξ±+Ξ²u+Ξ³v+Ξ΄uv)(\alpha+\beta u+\gamma v+\delta uv)(Ξ±+Ξ²u+Ξ³v+Ξ΄uv)- constacyclic codes overRRR. We give the structure of(Ξ±+Ξ²u+Ξ³v+Ξ΄uv)(\alpha+\beta u+\gamma v+\delta uv)(Ξ±+Ξ²u+Ξ³v+Ξ΄uv)- constacyclic codes overRRRand obtain self-orthogonal codes overFq\mathbb{F}_{q}Fq​by Gray map.
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
This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + Ξ½Z3 + Ο‰Z3 + Ξ½Ο‰Z3 where Ξ½2 = 1, Ο‰2 = Ο‰, Ξ½Ο‰ = Ξ½Ο‰, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.