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

Asian-European Journal of Mathematics

Quantum codes from cyclic codes over the ring Fp[u]/〈u3 − u〉

Tushar Bag, Ashish Kumar Upadhyay, Mohammad Ashraf, Ghulam Mohammad

Published Jul 27, 2018• 19 citations
Publisher

Abstract

In this paper, we study nonbinary quantum codes from cyclic codes over the ring [Formula: see text] for odd prime [Formula: see text]. We give the constructional properties of linear and cyclic codes over the ring [Formula: see text]. As an applications of these classes of codes, we obtain quantum codes over [Formula: see text] using self-orthogonal property. Moreover, we find some new nonbinary quantum codes with better parameters than the known quantum codes over [Formula: see text].

DOI
10.1142/s1793557120500084

Related papers

On quantum codes obtained from cyclic codes over 𝔽2 + u𝔽2 + u2𝔽2

Abhay Kumar Singh, Sukhamoy Pattanayak, Pratyush Kumar +1 more

In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.

Asian-European Journal of Mathematics•Feb 17, 2017•13 citations

New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq

Leilei Gao

Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.

International Journal of Discrete Mathematics•Jan 1, 2019•0 citations

New Non-Binary Quantum Codes from Cyclic Codes Over Product Rings

Tushar Bag, Hai Quang Dinh, Ashish Kumar Upadhyay +1 more

For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over Fpand FpSℓ, where Sℓ= Πi∈IℓRi, for Ri= (Fp[u])/(〈ui+1-u〉). For that, first we construct linear codes and a Gray map over Rℓ. Using this construction, we study cyclic codes over Rℓ, and then extend that over FpSℓ. We also give a Gray map over FpSℓ. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.

IEEE Communications Letters•Dec 13, 2019•25 citations

Construction of quantum codes from cyclic codes over Fp + vFp

Mohammad Ashraf, Ghulam Mohammad

In this paper, we study quantum error-correcting codes from cyclic codes over the ring R, where R is a non-chain extension of Fp. We give a method to obtain self-orthogonal codes over Fp as Gray images of linear and cyclic codes over the ring R. Finally, we derive the parameters of associated quantum error-correcting codes.

International Journal of Information and Coding Theory•Jan 1, 2015•34 citations

Self-dual cyclic and quantum codes over ℤ2 × (ℤ2 + uℤ2)

İsmail Aydoğdu, Taher Abualrub

In this paper, we introduce self-dual cyclic codes over the ring [Formula: see text]. We determine the conditions for any [Formula: see text]-cyclic code to be self-dual, that is, [Formula: see text]. Since the binary image of a self-orthogonal [Formula: see text]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over [Formula: see text]. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.

Discrete Mathematics Algorithms and Applications•Jun 19, 2019•8 citations

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

Xiaoshan Kai, Shixin Zhu

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

International Journal of Quantum Information•Mar 1, 2011•81 citations
In this paper, we study nonbinary quantum codes from cyclic codes over the ring [Formula: see text] for odd prime [Formula: see text]. We give the constructional properties of linear and cyclic codes over the ring [Formula: see text]. As an applications of these classes of codes, we obtain quantum codes over [Formula: see text] using self-orthogonal property. Moreover, we find some new nonbinary quantum codes with better parameters than the known quantum codes over [Formula: see text].
In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.
For any odd prime p, and a divisor ℓ of p, we consider Ie to be the set of all divisors of p - 1, which are less than or equal to ℓ. In this letter, we construct quantum codes from cyclic codes over Fpand FpSℓ, where Sℓ= Πi∈IℓRi, for Ri= (Fp[u])/(〈ui+1-u〉). For that, first we construct linear codes and a Gray map over Rℓ. Using this construction, we study cyclic codes over Rℓ, and then extend that over FpSℓ. We also give a Gray map over FpSℓ. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature.
QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER F4+uF4\mathbb{F}_4 + u\mathbb{F}_4F4​+uF4​