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

arXiv:2309.01983 · Advances in Mathematics of Communications

Quaternary conjucyclic codes with an application to EAQEC codes

Md Ajaharul Hossain, Ramakrishna Bandi

Published Mar 5, 2024• 1 citation
arXivHTMLPDF

Abstract

Conjucyclic codes are part of a family of codes that includes cyclic, constacyclic, and quasi-cyclic codes, among others. Despite their importance in quantum error correction, they have not received much attention in the literature. This paper focuses on additive conjucyclic (ACC) codes over $ \mathbb{F}_4 $ and investigates their properties. Specifically, we derive the duals of ACC codes by using a trace inner product and obtain the trace hull and its dimension. Also, we establish a necessary and sufficient condition for an additive code to have a complementary dual (ACD). Additionally, we identify a necessary condition for an additive conjucyclic complementary pair of codes over $ \mathbb{F}_4 $. Furthermore, we show that the trace code of an ACC code is cyclic and provide a condition for the trace code of an ACC code to be LCD. To demonstrate the practical application of our findings, we construct some good entanglement-assisted quantum error-correcting (EAQEC) codes using the trace code of ACC codes.

DOI
10.3934/amc.2024008

Related papers

Additive Conjucyclic Codes over $\F_{q^2}$: Trace Correspondence and Applications to Quantum Codes

Jingjie Lv, Ruihu Li, Ruihu Li +1 more

Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length $n$ correspond bijectively to $q$-ary linear cyclic codes of length $2n$, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $q$-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $q$-ary QECCs from additive conjucyclic codes, together with an illustrative example.

arXiv:2007.03963•Jul 8, 2020•0 citations

Additive conjucyclic codes over $ \mathbb{F}_{q^2} $: Trace correspondence and applications to quantum codes

Jingjie Lv, 廉贤, Ruihu Li +1 more

Additive conjucyclic codes over $ \mathbb{F}_{q^2} $ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $ \mathbb{F}_q $-linear isomorphism between $ \mathbb{F}_{q^2}^n $ and $ \mathbb{F}_q^{2n} $. This correspondence shows that additive conjucyclic codes of length $ n $ correspond bijectively to $ q $-ary linear cyclic codes of length $ 2n $, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $ \mathbb{F}_{q^2}^n $, which is shown to be compatible with the symplectic inner product on $ \mathbb{F}_q^{2n} $ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $ q $-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $ q $-ary QECCs from additive conjucyclic codes, together with an illustrative example.

AIMS Mathematics•Jan 1, 2026•0 citations

QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$

Rabia Zengin, Mehmet E. Köroğlu

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.

arXiv:2606.02137•Jun 1, 2026•0 citations

Additive conjucyclic codes over a class of Galois rings

Habibul Islam, Dipak K. Bhunia

Abstract As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except $${\mathbb {F}}_{q^2}$$ Fq2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings $$GR(2^r,2)$$ GR(2r,2) , where $$r\ge 2$$ r≥2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of lengthnover $$GR(2^r,2)$$ GR(2r,2) and the family of linear cyclic codes of length 2nover $${\mathbb {Z}}_{2^r}$$ Z2r . This correspondence helps to obtain additive conjucyclic codes over $$GR(2^r,2)$$ GR(2r,2) via known linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ Z2r . We prove that the trace dual $${\mathscr {C}}^{Tr}$$ CTr of an additive conjucyclic code $${\mathscr {C}}$$ C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over $${\mathbb {Z}}_{2^r}$$ Z2r contained in additive conjucyclic codes over $$GR(2^r,2)$$ GR(2r,2) . Last but not least, we explicitly derive the generator matrices for these codes.

Journal of Applied Mathematics and Computing•Dec 21, 2023•1 citation

Entanglement-assisted quantum error-correcting codes via quasi-cyclic codes with complementary duals

Hajime Matsui, Kakeru Kaneko

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.

Designs Codes and Cryptography•May 31, 2025•0 citations

Entanglement-assisted quantum error-correcting codes and asymmetric entanglement-assisted quantum error-correcting codes of length 5pt over 𝔽pm

Hai Quang Dinh, Bac Trong Nguyen, Van‐Nui Nguyen +1 more

Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.

International Journal of Modern Physics C•Feb 27, 2026•0 citations
Conjucyclic codes are part of a family of codes that includes cyclic, constacyclic, and quasi-cyclic codes, among others. Despite their importance in quantum error correction, they have not received much attention in the literature. This paper focuses on additive conjucyclic (ACC) codes over F4\mathbb{F}_4F4​ and investigates their properties. Specifically, we derive the duals of ACC codes by using a trace inner product and obtain the trace hull and its dimension. Also, we establish a necessary and sufficient condition for an additive code to have a complementary dual (ACD). Additionally, we identify a necessary condition for an additive conjucyclic complementary pair of codes over F4\mathbb{F}_4F4​. Furthermore, we show that the trace code of an ACC code is cyclic and provide a condition for the trace code of an ACC code to be LCD. To demonstrate the practical application of our findings, we construct some good entanglement-assisted quantum error-correcting (EAQEC) codes using the trace code of ACC codes.
Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length nnn correspond bijectively to qqq-ary linear cyclic codes of length 2n2n2n, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct qqq-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for qqq-ary QECCs from additive conjucyclic codes, together with an illustrative example.
Additive conjucyclic codes over $ \mathbb{F}_{q^2} $ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $ \mathbb{F}_q $-linear isomorphism between $ \mathbb{F}_{q^2}^n $ and $ \mathbb{F}_q^{2n} $. This correspondence shows that additive conjucyclic codes of length $ n $ correspond bijectively to $ q $-ary linear cyclic codes of length $ 2n $, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $ \mathbb{F}_{q^2}^n $, which is shown to be compatible with the symplectic inner product on $ \mathbb{F}_q^{2n} $ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $ q $-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $ q $-ary QECCs from additive conjucyclic codes, together with an illustrative example.
QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over F2×(F2+vF2)\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)F2​×(F2​+vF2​)
In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring F2×(F2+vF2)\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)F2​×(F2​+vF2​), where v2=vv^2 = vv2=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.
Abstract As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except Fq2{\mathbb {F}}_{q^2}Fq2​ Fq2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) , where r≥2r\ge 2r≥2 r≥2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of lengthnover GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) and the family of linear cyclic codes of length 2nover Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r . This correspondence helps to obtain additive conjucyclic codes over GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) via known linear cyclic codes over Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r . We prove that the trace dual CTr{\mathscr {C}}^{Tr}CTr CTr of an additive conjucyclic code C{\mathscr {C}}C C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r contained in additive conjucyclic codes over GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) . Last but not least, we explicitly derive the generator matrices for these codes.
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+xm-1+x^m−1+xm - 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.
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.