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

IEEE International Symposium on Information Theory (ISIT)

Dual-containing Alternant Codes for Applications in the Calderbank-Shor-Steane Construction

Sisi Miao, Christian Senger

Published Jul 12, 2021• 1 citation
Publisher

Abstract

Dual-containing classical error-correcting codes can be used in the Calderbank-Shor-Steane (CSS) construction for quantum error-correcting codes (QECCs). Dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes have been extensively studied in this regard. Research on the more general alternant codes is far less complete, despite the perspective of finding superior codes among them. In this paper, conditions for an alternant code to contain its Euclidean dual code are derived. The necessity of having dual-containing generalized Reed-Solomon (GRS) codes is shown and the structure of such codes is studied. An approach for constructing dual-containing alternant codes from BCH codes is proposed. In addition, a method for finding GRS codes over a given finite field with dual-containing alternant subfield-subcodes is presented. This method encompasses dual-containing BCH codes as a special case.

DOI
10.1109/isit45174.2021.9518127

Related papers

Some New Quantum BCH Codes over Finite Fields

Lijuan Xing, Zhuo Li

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

Entropy•Jun 3, 2021•4 citations

1 On the Construction of Nonbinary Quantum BCH Codes

Giuliano G. La Guardia

Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.

•Aug 16, 2016•39 citations

On the Construction of Nonbinary Quantum BCH Codes

Giuliano Gadioli La Guardia

Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.

arXiv:1212.5687•IEEE Transactions on Information Theory•Jan 31, 2014•61 citations

Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples

Zunaira Babar, Daryus Chandra, Hung Viet Nguyen +4 more

Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.

IEEE Communications Surveys & Tutorials•Jan 1, 2019•65 citations

On nonbinary quantum nonprimitive non-narrow-sense BCH codes

Giuliano G. La Guardia

Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. The first two ones are based on Calderbank-Shor-Steane (CSS) construction derived from two distinct nonprimitive BCH codes, not necessarily self-orthogonal. The third one is based on nonbinary Steane's enlargement of CSS codes applied to suitable sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction is derived from suitable sub-families of Hermitian self-orthogonal nonprimitive non-narrow-sense BCH codes. These constructions generate new families of quantum BCH codes whose parameters are better than the ones available in the literature.

International Symposium on Information Theory and its Applications•Oct 1, 2012•0 citations

Dual-Containing Determination Method for Non-Primitive BCH Codes over Finite Field

Wenjie Liu

Cyclotomic cosets play very important roles in classical and quantum error correction theory.In order to constructing quantum BCH(Bose Chaudhuri Hocquenghem) codes with CSS constructing method from classical BCH codes,it needs to determine whether a BCH code contains its dual.It proposed several properties of cyclotomic cosets.And according to these properties,an algorithm with polynomial time complexity was presented to determine whether a non-primitive BCH code over finite field contains its dual code,which can also be applied to nonnarrow sense primitive BCH codes.

Dianzi xuebao•Jan 1, 2010•0 citations
Dual-containing classical error-correcting codes can be used in the Calderbank-Shor-Steane (CSS) construction for quantum error-correcting codes (QECCs). Dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes have been extensively studied in this regard. Research on the more general alternant codes is far less complete, despite the perspective of finding superior codes among them. In this paper, conditions for an alternant code to contain its Euclidean dual code are derived. The necessity of having dual-containing generalized Reed-Solomon (GRS) codes is shown and the structure of such codes is studied. An approach for constructing dual-containing alternant codes from BCH codes is proposed. In addition, a method for finding GRS codes over a given finite field with dual-containing alternant subfield-subcodes is presented. This method encompasses dual-containing BCH codes as a special case.
Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.
Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. The first two ones are based on Calderbank-Shor-Steane (CSS) construction derived from two distinct nonprimitive BCH codes, not necessarily self-orthogonal. The third one is based on nonbinary Steane's enlargement of CSS codes applied to suitable sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction is derived from suitable sub-families of Hermitian self-orthogonal nonprimitive non-narrow-sense BCH codes. These constructions generate new families of quantum BCH codes whose parameters are better than the ones available in the literature.
Cyclotomic cosets play very important roles in classical and quantum error correction theory.In order to constructing quantum BCH(Bose Chaudhuri Hocquenghem) codes with CSS constructing method from classical BCH codes,it needs to determine whether a BCH code contains its dual.It proposed several properties of cyclotomic cosets.And according to these properties,an algorithm with polynomial time complexity was presented to determine whether a non-primitive BCH code over finite field contains its dual code,which can also be applied to nonnarrow sense primitive BCH codes.