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 Communications Letters

Quantum LDPC Codes from Balanced Incomplete Block Designs

Ivan B. Djordjević

Published May 1, 2008• 72 citations
Publisher

Abstract

We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes. The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9. Several examples of quantum LDPC codes from BIBDs from unity index, extended by addition of an all- ones column, are introduced as well. To improve the BER performance, we employed the method of removing the cycles

DOI
10.1109/lcomm.2008.080083

Related papers

A Rate Scalable Construction of Non-Homogeneous Quantum LDPC Codes of CSS Type Based on Balanced Incomplete Block Design

Yun-Jiang Wang, Zhuo-Yan Xiao, Xing-Yu Xiong +3 more

Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.

IEEE Communications Letters•Feb 25, 2021•4 citations

Design and Analysis of Quantum Dual-Containing CSS LDPC Codes based on Quasi-Dyadic Matrices

Alessio Baldelli, Marco Baldi, Massimo Battaglioni +2 more

Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.

cs.ITCross-listedarXiv:2605.03631•May 6, 2026•0 citations

Construction of Quantum LDPC Codes From Classical Row-Circulant QC-LDPCs

Zunaira Babar, Panagiotis Botsinis, Dimitrios Alanis +2 more

Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.

IEEE Communications Letters•Dec 10, 2015•18 citations

Quantum CSS LDPC Codes with Quasi-Dyadic Structure

Alessio Baldelli, Massimo Battaglioni, Paolo Santini

Quantum quasi-cyclic (QC) and quantum low-density parity-check (LDPC) codes have received significant attention due to their algebraic regularity and performance under low-complexity decoding. In this work, we explore a class of structured quantum codes based on reproducible matrices, which generalize known families such as cyclic, QC, and dyadic codes. Focusing on sparse quasi-dyadic (QD) structures, we investigate how their symmetries influence Tanner graph cycles. Moreover, we construct quantum LDPC codes within the Calderbank-Shor-Steane (CSS) framework, analyzing dual-containing configurations, and assess their error rates.

International Symposium on Turbo Codes and Iterative Information Processing•Aug 18, 2025•2 citations

Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

Alessio Baldelli, Massimo Battaglioni, Jonathan Mandelbaum +2 more

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.

cs.ITCross-listedarXiv:2601.08636•IEEE International Symposium on Information Theory (ISIT)•Jan 14, 2026•0 citations

Construction of well-structured quasi-cyclic low-density parity check codes

B. Honary, Abdolhosein Moinian, Bassem Ammar

Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.

IEE Proceedings - Communications•Jan 1, 2005•10 citations
We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes. The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9. Several examples of quantum LDPC codes from BIBDs from unity index, extended by addition of an all- ones column, are introduced as well. To improve the BER performance, we employed the method of removing the cycles
Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.
Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.
Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.
Quantum quasi-cyclic (QC) and quantum low-density parity-check (LDPC) codes have received significant attention due to their algebraic regularity and performance under low-complexity decoding. In this work, we explore a class of structured quantum codes based on reproducible matrices, which generalize known families such as cyclic, QC, and dyadic codes. Focusing on sparse quasi-dyadic (QD) structures, we investigate how their symmetries influence Tanner graph cycles. Moreover, we construct quantum LDPC codes within the Calderbank-Shor-Steane (CSS) framework, analyzing dual-containing configurations, and assess their error rates.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.