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

Construction of Girth-Eight QC-LDPC Codes from Greatest Common Divisor

Guohua Zhang, Rong Xia Sun, Xinmei Wang

Published Jan 2, 2013• 47 citations
Publisher

Abstract

For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.

DOI
10.1109/lcomm.2012.122012.122292

Related papers

Relation Between GCD Constraint and Full-Length Row-Multiplier QC-LDPC Codes With Girth Eight

Guohua Zhang, Yulin Hu, Yi Fang +1 more

The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of (J,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any J and any L.

IEEE Communications Letters•Jul 12, 2021•12 citations

Automatic Verification of GCD Constraint for Construction of Girth-Eight QC-LDPC Codes

Guohua Zhang, Yi Fang, Yuanhua Liu

As a general method to yield constructions for quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight, the greatest-common-divisor (GCD) framework heavily relies on verifying a type of inequalities, referred to as GCD constraints. An algorithm is developed in this letter to automatically verify GCD constraints without conducting any manual analysis, by bounding from above the GCD of a fixed integer and an integer in the form of linear function. As an application of the algorithm, a set of novel constructions based on GCD framework is proposed. From these new constructions, four novel bounds on the size of circulant permutation matrices (CPMs) are formulated, such that girth-eight QC-LDPC codes always exist for any CPM size greater than or equal to the bounds. The new bound for column weight of 6 slightly improves the existing best one, and those for column weights from 7 to 9 significantly strengthen the state-of-the-art ones by decreasing from essentially a cubic or biquadratic power of row weight to a quadratic power of row weight.

IEEE Communications Letters•Jun 28, 2019•11 citations

Constructions of Girth-Eight QC-LDPC Codes with Dual-Diagonal Structure Based on GCD Framework

Guohua Zhang, Haiyang Liu, Mengjuan Lou +1 more

A novel class of quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly proposed without 4-cycles and 6-cycles via the greatest-common-divisor (GCD) framework. From the new codes, a novel type of girth-eight QC-LDPC codes with a dual-diagonal (DD) structure is presented, in which the single undetermined entry in the associated exponent matrix can be determined by simple formulae. The proposed DD codes have similar or better performance compared with DD codes obtained from computer search. Moreover, a lower bound on circulant sizes which guarantees the existence of qualified undetermined entries is also provided.

International Conference on Wireless Communications and Signal Processing (WCSP)•Nov 2, 2023•5 citations

QC-LDPC Codes with Girth Eight Based on Independent Row-Column Mapping Sequence

Lei Wang, Xing Zhang, Feng Yu +2 more

Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.

IEEE Communications Letters•Oct 10, 2013•6 citations

Constructions of Type-II QC-LDPC Codes With Girth Eight from Sidon Sequence

Guohua Zhang, Yulin Hu, Yi Fang +1 more

In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.

IEEE Transactions on Communications•Feb 25, 2019•22 citations

On the Construction of Multitype Quasi-Cyclic Low-Density Parity-Check Codes With Different Girth and Length

Farzaneh Abedi, Mohammad Gholami

Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth$g_{\max }$of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth$g_{\max }$, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth$g$,$g\le g_{\max }$, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.

IEEE Access•Jan 1, 2021•2 citations
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of (J,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any J and any L.
As a general method to yield constructions for quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight, the greatest-common-divisor (GCD) framework heavily relies on verifying a type of inequalities, referred to as GCD constraints. An algorithm is developed in this letter to automatically verify GCD constraints without conducting any manual analysis, by bounding from above the GCD of a fixed integer and an integer in the form of linear function. As an application of the algorithm, a set of novel constructions based on GCD framework is proposed. From these new constructions, four novel bounds on the size of circulant permutation matrices (CPMs) are formulated, such that girth-eight QC-LDPC codes always exist for any CPM size greater than or equal to the bounds. The new bound for column weight of 6 slightly improves the existing best one, and those for column weights from 7 to 9 significantly strengthen the state-of-the-art ones by decreasing from essentially a cubic or biquadratic power of row weight to a quadratic power of row weight.
A novel class of quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly proposed without 4-cycles and 6-cycles via the greatest-common-divisor (GCD) framework. From the new codes, a novel type of girth-eight QC-LDPC codes with a dual-diagonal (DD) structure is presented, in which the single undetermined entry in the associated exponent matrix can be determined by simple formulae. The proposed DD codes have similar or better performance compared with DD codes obtained from computer search. Moreover, a lower bound on circulant sizes which guarantees the existence of qualified undetermined entries is also provided.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.
Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girthgmax⁡g_{\max }gmax​of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girthgmax⁡g_{\max }gmax​, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girthggg,g≤gmax⁡g\le g_{\max }g≤gmax​, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.