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:2605.23447

AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$

Manasa J, Divya Acharya, Prasanna Poojary

Published May 22, 2026• 0 citations
arXivHTMLPDF

Abstract

This paper provides a comprehensive analysis of almost maximum distance separable (AMDS) constacyclic codes of length $4p^ς$ over the finite field $\mathbb{F}_{p^m}$, where $p$ is an odd prime number. Furthermore, it introduces the construction of quantum AMDS (qAMDS) codes, drawing on the principles of the Calderbank-Shor-Steane (CSS) framework, which enhances their applicability in quantum error correction. This work aims to deepen the understanding of these codes and their potential uses in both classical and quantum computing environments.

DOI
10.48550/arxiv.2605.23447

Related papers

Quantum AMDS Negacyclic Codes of Length 2 p s Over F p m

Hai Quang Dinh, Bac Trong Nguyen, Nui V Nguyen +2 more

In this article, we investigate all almost maximum distance separable (briefly, AMDS) negacyclic codes of length 2psover the finite field Fpm, wherepis an odd prime. A necessary and sufficient condition for a code to be an AMDS and NMDS negacyclic code of length 2psover Fpmis also provided. In addition, we construct quantum AMDS (qAMDS) codes using both the Calderbank-Shor-Steane (CSS) and Hermitian constructions. These newly constructed codes have different parameters compared to previous constructions. Moreover, our qAMDS codes are better than all known quantum error correction (QEC) codes.

IEEE Access•Aug 4, 2025•0 citations

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2ps Over F pm

Hai Quang Dinh, Bac Trong Nguyen, Woraphon Yamaka

Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.

IEEE Access•Jan 1, 2020•16 citations

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

Lanqiang Li, Ziwen Cao, Tingting Wu +1 more

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.

arXiv:2406.13943•Jun 20, 2024•0 citations

New quantum codes from skew constacyclic codes

Ram Krishna Verma, Om Prakash, Ashutosh Singh +1 more

For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

Advances in Mathematics of Communications•Aug 19, 2021•19 citations

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and application to quantum codes

Yongsheng Tang, Ting Yao, Shixin Zhu +1 more

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

•Dec 6, 2017•0 citations

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and its application to quantum codes

Yongsheng Tang, Ting Yao, Shixin Zhu +1 more

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

arXiv:1712.02081•Dec 6, 2017•0 citations
AMDS and quantum AMDS Constacyclic codes of length 4pς4p^ς4pς over Fpm\mathbb{F}_{{p}^{m}}Fpm​
This paper provides a comprehensive analysis of almost maximum distance separable (AMDS) constacyclic codes of length 4pς4p^ς4pς over the finite field Fpm\mathbb{F}_{p^m}Fpm​, where ppp is an odd prime number. Furthermore, it introduces the construction of quantum AMDS (qAMDS) codes, drawing on the principles of the Calderbank-Shor-Steane (CSS) framework, which enhances their applicability in quantum error correction. This work aims to deepen the understanding of these codes and their potential uses in both classical and quantum computing environments.
In this article, we investigate all almost maximum distance separable (briefly, AMDS) negacyclic codes of length 2psover the finite field Fpm, wherepis an odd prime. A necessary and sufficient condition for a code to be an AMDS and NMDS negacyclic code of length 2psover Fpmis also provided. In addition, we construct quantum AMDS (qAMDS) codes using both the Calderbank-Shor-Steane (CSS) and Hermitian constructions. These newly constructed codes have different parameters compared to previous constructions. Moreover, our qAMDS codes are better than all known quantum error correction (QEC) codes.
Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.
New QEC codes and EAQEC codes from repeated-root cyclic codes of length 2rps2^rp^s2rps
Let ppp be an odd prime and r,s,mr,s,mr,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 2rps2^rp^s2rps over the finite field Fpm\mathbb{F}_{p^m}Fpm​. 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 2rps2^rp^s2rps, 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 $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.
A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and application to quantum codes
We introduce a Gray map from F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ to F22m\mathbb{F}_{2}^{2m}F22m​ and study (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m,\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},F2m​+uF2m​, where u2=0.u^{2}=0.u2=0. It is proved that the image of a (1+u)(1+u)(1+u)-constacyclic code length nnn over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ under the Gray map is a distance-invariant quasi-cyclic code of index mmm and length 2mn2mn2mn over F2.\mathbb{F}_{2}.F2​. We also prove that every code of length 2mn2mn2mn which is the Gray image of cyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ of length nnn is permutation equivalent to a binary quasi-cyclic code of index m.m.m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m.\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.F2m​+uF2m​.
A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and its application to quantum codes
We introduce a Gray map from F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ to F22m\mathbb{F}_{2}^{2m}F22m​ and study (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m,\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},F2m​+uF2m​, where u2=0.u^{2}=0.u2=0. It is proved that the image of a (1+u)(1+u)(1+u)-constacyclic code length nnn over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ under the Gray map is a distance-invariant quasi-cyclic code of index mmm and length 2mn2mn2mn over F2.\mathbb{F}_{2}.F2​. We also prove that every code of length 2mn2mn2mn which is the Gray image of cyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ of length nnn is permutation equivalent to a binary quasi-cyclic code of index m.m.m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1+u)(1+u)(1+u)-constacyclic codes over F2m+uF2m.\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.F2m​+uF2m​.