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

Journal of Discrete Mathematical Sciences and Cryptography

ϒ – Constacyclic codes over a Frobenius ring and their application to find novel quantum codes

Pooja Soni, Manju Pruthi, Arun Kumar Yadav

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this work, we use the Hermitian technique to find novel Quantum error correcting codes (𝒬E𝒞𝒞’s) by Constacyclic codes (CCc’s) over the ring ℛ = ℱp2 [𝓊, 𝓋]/(𝓊3 – ω2 𝓊, 𝓋3 – ω2 𝓋), where p is a non-even prime power and ℱ*p 2 = 〈ω〉. Firstly, we express the ring in terms of “pairwise orthogonal Idempotent” elements & analyze “linear codes.” We investigate ring structure and establish a one-one, onto and distance preserving Gray map, ∅ ∶ ℛ𝓃→ℱ5𝓃 p2. We provide the units of ℛ, to analyze CCc’s & their generators. We also give a condition for this code to contain its Hermitian dual over ℛ. At last, we derive and tabulate some novel and efficient 𝒬E𝒞𝒞’s.

DOI
10.47974/jdmsc-2415

Related papers

Quantum Codes Obtained From Constacyclic Codes Over a Family of Finite Rings Fp[u₁, u₂, …, us]

Hai Quang Dinh, Tushar Bag, Sachin Pathak +2 more

In this article, we construct some MDS quantum error-correcting codes (QECCs) from classes of constacyclic codes over Rs= Fp+ u1Fp+ ··· + usFp, ui2= ui, uiuj= ujui= 0, for odd prime p and i, j = 1, 2,⋯, s, i ≠ j. Many QECCs with improved parameters than the existing ones in some of the earlier papers are provided. We present a set of idempotent generators of the ring Rs, and using that we define linear codes, determine all units, and study constacyclic codes over this ring. Among others, we study dual containing constacyclic codes over Rs and construct (non-binary) QECCs. An algorithm to construct QECCs from dual containing constacyclic codes over Rs is obtained that can provide many quantum codes.

IEEE Access•Jan 1, 2020•12 citations

Constacyclic Codes Over 𝑭𝒒[𝒖] /〈𝒖𝟑 = 𝟎〉 and Their Application of Constructing Quantum Codes

Zineb Hebbache

Let 𝑅 = 𝐹𝑞+u𝐹𝑞+𝑢^2𝐹𝑞, 𝑢^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 𝐹𝑞 by using the Gray map from 𝑅𝑛 to 𝐹𝑞^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.

International Journal of Informatics and Applied Mathematics•Nov 11, 2022•0 citations

ON QUANTUM CODES CONSTRUCTION FROM CONSTACYCLIC CODES OVER THE RING I_q[u,v] / <u^2-a^2, v^2-a^2, uv-vu>

Shakir Ali, Pushpendra Sharma

This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring $\mathfrak{S}$, which is defined as $\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangle$, where $\mathfrak{I}_q$ is a finite field of $q$ elements, $\alpha$ be the nonzero elements of the field $\mathfrak{I}_q$ and $q$ is a power of an odd prime $p$ such that $q=p^m, ~\textup{for}~ m \ge 1$. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring $\mathfrak{S}$ into a direct sum of constacyclic codes over $\mathfrak{I}_q$. We construct new and better quantum error-correcting codes over the ring $\mathfrak{S}$ (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).

Journal of the Indonesian Mathematical Society•Aug 21, 2024•0 citations

Quantum codes from a class of constacyclic codes over finite commutative rings

Hai Quang Dinh, Tushar Bag, Ashish Kumar Upadhyay +3 more

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Journal of Algebra and Its Applications•Oct 31, 2019•25 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

Constacyclic codes over 𝔽 q 𝒮𝒯 and their applications

Amal S. Alali, Mohd Azmi, Nadeem ur Rehman +1 more

Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.

AKCE International Journal of Graphs and Combinatorics•Sep 23, 2024•0 citations
In this work, we use the Hermitian technique to find novel Quantum error correcting codes (𝒬E𝒞𝒞’s) by Constacyclic codes (CCc’s) over the ring ℛ = ℱp2 [𝓊, 𝓋]/(𝓊3 – ω2 𝓊, 𝓋3 – ω2 𝓋), where p is a non-even prime power and ℱ*p 2 = 〈ω〉. Firstly, we express the ring in terms of “pairwise orthogonal Idempotent” elements & analyze “linear codes.” We investigate ring structure and establish a one-one, onto and distance preserving Gray map, ∅ ∶ ℛ𝓃→ℱ5𝓃 p2. We provide the units of ℛ, to analyze CCc’s & their generators. We also give a condition for this code to contain its Hermitian dual over ℛ. At last, we derive and tabulate some novel and efficient 𝒬E𝒞𝒞’s.
In this article, we construct some MDS quantum error-correcting codes (QECCs) from classes of constacyclic codes over Rs= Fp+ u1Fp+ ··· + usFp, ui2= ui, uiuj= ujui= 0, for odd prime p and i, j = 1, 2,⋯, s, i ≠ j. Many QECCs with improved parameters than the existing ones in some of the earlier papers are provided. We present a set of idempotent generators of the ring Rs, and using that we define linear codes, determine all units, and study constacyclic codes over this ring. Among others, we study dual containing constacyclic codes over Rs and construct (non-binary) QECCs. An algorithm to construct QECCs from dual containing constacyclic codes over Rs is obtained that can provide many quantum codes.
This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S\mathfrak{S}S, which is defined as S=Iq[u,v]/⟨u2−α2, v2−α2, uv−vu⟩\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangleS=Iq​[u,v]/⟨u2−α2, v2−α2, uv−vu⟩, where Iq\mathfrak{I}_qIq​ is a finite field of qqq elements, α\alphaα be the nonzero elements of the field Iq\mathfrak{I}_qIq​ and qqq is a power of an odd prime ppp such that q=pm, for m≥1q=p^m, ~\textup{for}~ m \ge 1q=pm, for m≥1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S\mathfrak{S}S into a direct sum of constacyclic codes over Iq\mathfrak{I}_qIq​. We construct new and better quantum error-correcting codes over the ring S\mathfrak{S}S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).
Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].
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.
Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.