Journal of Algebra Combinatorics Discrete Structures and Applications

EAQEC codes from polycyclic codes over the ring $R_l$

R. Gokul, Karthick Gowdhaman, Mahalakshmi Jothi Ramanujam, Cruz Mohan, Chinnapillai Durairajan

Published May 6, 2026• 0 citations

Abstract

In this paper, we study polycyclic codes over the ring $ R_l = \frac{\mathbb{F}_q[w]}{\langle w^l - 1 \rangle} $ with $ q = p^k $ where $ k $ is a positive integer and $ p $ is an odd prime. We explore LCD annihilator, self-dual, self-orthogonal codes over $ R_l $ and polycyclic codes over $ R_l $. Moreover, we provide a structure of entanglement-assisted quantum error-correcting codes based on the developed polycyclic codes through the inclusion of dual contained conditions. Subsequently, some LCD hull code-derived entanglement-assisted quantum error-correcting code examples are presented.Accepted: 16 March 2026

DOI: 10.13069/jacodesmath.v13i2.390

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Journal of Algebra Combinatorics Discrete Structures and Applications

EAQEC codes from polycyclic codes over the ring $R_l$ EAQEC codes from polycyclic codes over the ring $R_l$

R. Gokul, Karthick Gowdhaman, Mahalakshmi Jothi Ramanujam, Cruz Mohan, Chinnapillai Durairajan

Published May 6, 2026• 0 citations

Publisher

Abstract

In this paper, we study polycyclic codes over the ring ( R_l = \frac{\mathbb{F}_q[w]}{\langle w^l - 1 \rangle} ) with ( q = p^k ) where ( k ) is a positive integer and ( p ) is an odd prime. We explore LCD annihilator, self-dual, self-orthogonal codes over ( R_l ) and polycyclic codes over ( R_l ). Moreover, we provide a structure of entanglement-assisted quantum error-correcting codes based on the developed polycyclic codes through the inclusion of dual contained conditions. Subsequently, some LCD hull code-derived entanglement-assisted quantum error-correcting code examples are presented.Accepted: 16 March 2026

DOI: 10.13069/jacodesmath.v13i2.390

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Polycyclic Codes and Their Application in Constructing AEAQECCs

Juan Li, Yunbo Tian, Fanghui Ma

In this article, we study polycyclic codes over the ring R=Fq[v]/⟨v2−1⟩, where q=pm with p being an odd prime. First, we introduce polycyclic codes and sequential codes over R, and characterize the structural properties of these polycyclic codes. Next, we analyze the Euclidean dual codes, annihilator dual codes, annihilator self-orthogonal codes, and annihilator linear complementary dual (LCD) codes associated with this family of codes. Finally, some asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) are constructed from polycyclic codes over R. Moreover, the parameters of our AEAQECCs are new in the existing literature.In this article, we study polycyclic codes over the ring R=Fq[v]/⟨v2−1⟩, where q=pm with p being an odd prime. First, we introduce polycyclic codes and sequential codes over R, and characterize the structural properties of these polycyclic codes. Next, we analyze the Euclidean dual codes, annihilator dual codes, annihilator self-orthogonal codes, and annihilator linear complementary dual (LCD) codes associated with this family of codes. Finally, some asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) are constructed from polycyclic codes over R. Moreover, the parameters of our AEAQECCs are new in the existing literature.

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In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$ , where $v^2 = v$ . Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$, where $v^2 = v$. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

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Abstract It is known that entanglement-assisted quantum error-correcting codes (EAQECCs), a type of quantum error correction codes, can be easily constructed using linear codes that satisfy the property called linear complementary duals (LCD). For quasi-cyclic (QC) codes, which are a class of linear codes, we have already published the methods for constructing the codes with properties such as self-orthogonality, self-duality and reversibility according to the prime-factor decomposition of $-1+x^m$ - 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.Abstract It is known that entanglement-assisted quantum error-correcting codes (EAQECCs), a type of quantum error correction codes, can be easily constructed using linear codes that satisfy the property called linear complementary duals (LCD). For quasi-cyclic (QC) codes, which are a class of linear codes, we have already published the methods for constructing the codes with properties such as self-orthogonality, self-duality and reversibility according to the prime-factor decomposition of $$-1+x^m$$ - 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.

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New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$ New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$

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Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$ . Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$ , which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.

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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 94B99Abstract 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 ProcessingJul 18, 20246 citations

EAQEC codes from polycyclic codes over the ring

R_l

QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over

\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)

In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring

\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)

, where

v^2 = v

. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

-1+x^m

- 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.

New QEC codes and EAQEC codes from repeated-root cyclic codes of length

2^rp^s

Let

p

be an odd prime and

r,s,m

be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of

2^rp^s

over the finite field

\mathbb{F}_{p^m}

. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length

2^rp^s

, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.

For an odd prime power q and

{\mathcal{R}}_{7}=\mathbb{F}_{q}+u \mathbb{F}_{q}+

v \mathbb{F}_{q}+w \mathbb{F}_{q}+z \mathbb{F}_{q}+u^{2} \mathbb{F}_{q}+w z \mathbb{F}_{q}

. Through this work we seek to construct new quantum codes and LCD codes by means of cyclic codes built on the ring

{\mathcal{R}}_{7}

. By using cyclic codes containing their duals, we give quantum codes, we also give a method for the construction of LCD Codes over

{\mathcal{R}}_{7}

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