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

Information Theory and Applications Workshop (ITA)

Cyclic and quasi-cyclic LDPC codes: New developments

Qin Huang, Qiuju Diao, Shu Lin, Khaled Abdel-Ghaffar

Published Feb 1, 2011• 11 citations
Publisher

Abstract

This paper presents a technique to decompose a cyclic code given by a parity-check matrix in circulant form into descendant cyclic and quasi-cyclic codes of various length and rates. Based on this technique, cyclic finite geometry (FG) LDPC codes are decomposed into a large class of cyclic FG-LDPC codes and a large class of quasi-cyclic FG-LDPC codes.

DOI
10.1109/ita.2011.5743581

Related papers

Circulant decomposition: Cyclic, quasi-cyclic and LDPC codes

Qin Huang, Qiuju Diao, Shu Lin

This paper shows that a cyclic code can be put into quasi-cyclic form by decomposing a circular parity-check matrix through column and row permutations. Such a decomposition of a circular parity-check matrix of a cyclic code produces a group of shorter cyclic or quasi-cyclic codes and leads to a new method for constructing long cyclic codes from short cyclic codes. Also in this paper, new classes of cyclic and quasi-cyclic LDPC codes are derived from cyclic Euclidean geometry LDPC codes by decomposing their circular parity-check matrices. These new LDPC codes perform well and enlarge the repertoire of cyclic and quasi-cyclic LDPC codes.

International Symposium on Information Theory and its Applications•Oct 1, 2010•5 citations

Cyclic and Quasi-Cyclic LDPC Codes on Constrained Parity-Check Matrices and Their Trapping Sets

Qin Huang, Qiuju Diao, Shu Lin +1 more

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.

IEEE Transactions on Information Theory•Jan 31, 2012•108 citations

Cyclic and Quasi-Cyclic LDPC Codes on Row and Column Constrained Parity-Check Matrices and Their Trapping Sets

Qin Huang, Qiuju Diao, Shu Lin +1 more

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and quasi-cyclic LDPC codes with large minimum weights are shown to have no harmful trapping sets with size smaller than their minimum weights. Consequently, their performance error-floors are dominated by their minimum weights.

arXiv:1012.3201•Dec 15, 2010•12 citations

A construction of the high-rate regular quasi-cyclic LDPC codes

Meng Qi, Jia Xiang Zhao, Wei Min Xu +1 more

In this paper, a scheme to construct the high-rate regular quasi-cyclic low-density parity-check (QC-LDPC) codes is developed based on the finite geometry low-density parity-check (LDPC) codes. To achieve this, we first decompose the EG-LDPC code into a set of the block submatrices with each of them being cyclic. Utilizing the decomposed structure, we then construct an auxiliary matrix to facilitate us for identifying and eliminating 6-cycles of the block submatrices. In the simulation, seven regular QC-LDPC codes with code rates being 4/5, 7/8, 10/11, 11/12, and 12/13 of moderate code lengths are constructed with this scheme. The performances of these codes demonstrate the effectiveness of the proposed scheme.

EURASIP Journal on Wireless Communications and Networking•Jan 17, 2019•3 citations

Finite-Geometry LDPC Codes

W.E. Ryan, Shu Lin

Finite geometries, such as Euclidean and projective geometries, are powerful mathematical tools for constructing error-control codes. In the 1960s and 1970s, finite geometries were successfully used to construct many classes of easily implementable majority-logic decodable codes. In 2000, Kou, Lin, and Fossorier [1–3] showed that finite geometries can also be used to construct LDPC codes that perform well and close to the Shannon theoretical limit with iterative decoding based on belief propagation. These codes are called finite-geometry (FG) -LDPC codes. FG-LDPC codes form the first class of LDPC codes that are constructed algebraically. Since 2000, there have been many major developments in construction of LDPC codes based on various structural properties of finite geometries [4–18]. In this chapter, we put together all the major constructions of LDPC codes based on finite geometries under a unified frame. We begin with code constructions based on Euclidean geometries and then go on to discuss those based on projective geometries. Construction of LDPC Codes Based on Lines of Euclidean Geometries This section presents a class of cyclic LDPC codes and a class of quasi-cyclic (QC) LDPC codes constructed using lines of Euclidean geometries. Before we present the constructions of these two classes of Euclidean-geometry (EG)-LDPC codes, we recall some fundamental structural properties of a Euclidean geometry that have been discussed in Chapter 2.

Cambridge University Press eBooks•Sep 17, 2009•1 citation

Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields

Li Guo Zhang, Shu Lin, Khaled Abdel-Ghaffar +2 more

A new class of quasi-cyclic LDPC codes whose parity-check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented. This class of codes contains several known classes of algebraic quasi-cyclic LDPC codes as subclasses. Experimental results show that the codes constructed perform very well over the AWGN channel when decoded with iterative decoding based on belief propagation. This class of new QC-LDPC codes contains a subclass of codes which have large minimum distances. Combinatorial expressions for the ranks of the parity-check matrices of a subclass of codes constructed based on fields of characteristic two are given.

IEEE Transactions on Communications•Jun 28, 2011•34 citations
This paper presents a technique to decompose a cyclic code given by a parity-check matrix in circulant form into descendant cyclic and quasi-cyclic codes of various length and rates. Based on this technique, cyclic finite geometry (FG) LDPC codes are decomposed into a large class of cyclic FG-LDPC codes and a large class of quasi-cyclic FG-LDPC codes.
This paper shows that a cyclic code can be put into quasi-cyclic form by decomposing a circular parity-check matrix through column and row permutations. Such a decomposition of a circular parity-check matrix of a cyclic code produces a group of shorter cyclic or quasi-cyclic codes and leads to a new method for constructing long cyclic codes from short cyclic codes. Also in this paper, new classes of cyclic and quasi-cyclic LDPC codes are derived from cyclic Euclidean geometry LDPC codes by decomposing their circular parity-check matrices. These new LDPC codes perform well and enlarge the repertoire of cyclic and quasi-cyclic LDPC codes.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and quasi-cyclic LDPC codes with large minimum weights are shown to have no harmful trapping sets with size smaller than their minimum weights. Consequently, their performance error-floors are dominated by their minimum weights.
In this paper, a scheme to construct the high-rate regular quasi-cyclic low-density parity-check (QC-LDPC) codes is developed based on the finite geometry low-density parity-check (LDPC) codes. To achieve this, we first decompose the EG-LDPC code into a set of the block submatrices with each of them being cyclic. Utilizing the decomposed structure, we then construct an auxiliary matrix to facilitate us for identifying and eliminating 6-cycles of the block submatrices. In the simulation, seven regular QC-LDPC codes with code rates being 4/5, 7/8, 10/11, 11/12, and 12/13 of moderate code lengths are constructed with this scheme. The performances of these codes demonstrate the effectiveness of the proposed scheme.
Finite geometries, such as Euclidean and projective geometries, are powerful mathematical tools for constructing error-control codes. In the 1960s and 1970s, finite geometries were successfully used to construct many classes of easily implementable majority-logic decodable codes. In 2000, Kou, Lin, and Fossorier [1–3] showed that finite geometries can also be used to construct LDPC codes that perform well and close to the Shannon theoretical limit with iterative decoding based on belief propagation. These codes are called finite-geometry (FG) -LDPC codes. FG-LDPC codes form the first class of LDPC codes that are constructed algebraically. Since 2000, there have been many major developments in construction of LDPC codes based on various structural properties of finite geometries [4–18]. In this chapter, we put together all the major constructions of LDPC codes based on finite geometries under a unified frame. We begin with code constructions based on Euclidean geometries and then go on to discuss those based on projective geometries. Construction of LDPC Codes Based on Lines of Euclidean Geometries This section presents a class of cyclic LDPC codes and a class of quasi-cyclic (QC) LDPC codes constructed using lines of Euclidean geometries. Before we present the constructions of these two classes of Euclidean-geometry (EG)-LDPC codes, we recall some fundamental structural properties of a Euclidean geometry that have been discussed in Chapter 2.
A new class of quasi-cyclic LDPC codes whose parity-check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented. This class of codes contains several known classes of algebraic quasi-cyclic LDPC codes as subclasses. Experimental results show that the codes constructed perform very well over the AWGN channel when decoded with iterative decoding based on belief propagation. This class of new QC-LDPC codes contains a subclass of codes which have large minimum distances. Combinatorial expressions for the ranks of the parity-check matrices of a subclass of codes constructed based on fields of characteristic two are given.