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 Transactions on Information Theory

Structural Analysis of Generalized Quasi-Cyclic LDPC Codes: Rank, Design, and Generator Matrices

Roxana Smarandache, David G. M. Mitchell, Anthony Gómez-Fonseca

Published Oct 27, 2025• 3 citations
Publisher

Abstract

Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.

DOI
10.1109/tit.2025.3625539

Related papers

On Generalized LDPC Codes for 5G Ultra Reliable Communication

Yanfang Liu, Pablo M. Olmos, David G. M. Mitchell

Generalized low-density parity-check (GLDPC) codes, where single parity-check (SPC) constraint nodes are replaced with generalized constraint (GC) nodes, are a promising class of codes for low latency communication. In this paper, a practical construction of quasi-cyclic (QC) GLDPC codes is proposed, where the proportion of generalized constraints is determined by an asymptotic analysis. We analyze the message passing process and complexity of a GLDPC code over the additive white gaussian noise (AWGN) channel and present a constraint-to-variable update rule based on the specific codewords of the component code. The block error rate (BLER) performance of the GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G Ultra Reliable Communication (URC) regime over an additive white gaussian noise (AWGN) channel with quadrature PSK (QPSK) modulation.

arXiv:1910.06574•IEEE Information Theory Workshop (ITW)•Nov 1, 2018•9 citations

Rank Analysis of Parity-Check Matrices for Quasi-Cyclic LDPC Codes

Po‐Chun Yang, Chung-Hsuan Wang, Chi-Chao Chao

Quasi-cyclic low-density parity-check (QC-LDPC) codes are an important class of LDPC codes which can be encoded and decoded with low complexity and suitable for many applications. As the code dimension, which describes the number of protected information bits, is equal to the code length minus the rank of the parity-check matrix and the parity-check matrix for QC-LDPC codes is usually not full-rank, determining the rank of the parity-check matrix is of essential importance. In this paper, we study the rank of the parity-check matrix for QC-LDPC codes based on the associated polynomials for circulant matrices. A formula for the rank of the parity-check matrix with only one row-block is first derived. We then extend the result to matrices with two, three, or more row-blocks. Some bounds are also presented for matrices with arbitrary numbers of row-blocks. Furthermore, the exact rank is determined for a class of algebraically constructed parity-check matrices.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2018•5 citations

Using Minors to Construct Generator Matrices for Quasi-Cyclic LDPC Codes

Roxana Smarandache, Anthony Gómez-Fonseca, David G. M. Mitchell

This paper gives a simple method to construct generator matrices with polynomial entries (and hence offers an alternative encoding method to the one commonly used) for all quasi-cyclic low-density parity-check (QC-LDPC) codes, even for those that are rank deficient. The approach is based on constructing a set of codewords with the desired total rank by using minors of the parity-check matrix. We exemplify the method on several well-known and standard codes. Moreover, we explore the connections between the minors of the parity-check matrix and the known upper bound on minimum distance and provide a method to compute the rank of any parity-check matrix representing a QC-LDPC code, and hence the dimension of the code, by using the minors of the corresponding polynomial parity-check matrix.

IEEE International Symposium on Information Theory (ISIT)•Jun 26, 2022•5 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

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

Design of Quasi-Cyclic Cycle LDPC Codes over GF(q)

HU Shu-kai, Chao Chen, Rong Xia Sun +1 more

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.

IEICE Transactions on Communications•Jan 1, 2012•0 citations
Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.
Generalized low-density parity-check (GLDPC) codes, where single parity-check (SPC) constraint nodes are replaced with generalized constraint (GC) nodes, are a promising class of codes for low latency communication. In this paper, a practical construction of quasi-cyclic (QC) GLDPC codes is proposed, where the proportion of generalized constraints is determined by an asymptotic analysis. We analyze the message passing process and complexity of a GLDPC code over the additive white gaussian noise (AWGN) channel and present a constraint-to-variable update rule based on the specific codewords of the component code. The block error rate (BLER) performance of the GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G Ultra Reliable Communication (URC) regime over an additive white gaussian noise (AWGN) channel with quadrature PSK (QPSK) modulation.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are an important class of LDPC codes which can be encoded and decoded with low complexity and suitable for many applications. As the code dimension, which describes the number of protected information bits, is equal to the code length minus the rank of the parity-check matrix and the parity-check matrix for QC-LDPC codes is usually not full-rank, determining the rank of the parity-check matrix is of essential importance. In this paper, we study the rank of the parity-check matrix for QC-LDPC codes based on the associated polynomials for circulant matrices. A formula for the rank of the parity-check matrix with only one row-block is first derived. We then extend the result to matrices with two, three, or more row-blocks. Some bounds are also presented for matrices with arbitrary numbers of row-blocks. Furthermore, the exact rank is determined for a class of algebraically constructed parity-check matrices.
This paper gives a simple method to construct generator matrices with polynomial entries (and hence offers an alternative encoding method to the one commonly used) for all quasi-cyclic low-density parity-check (QC-LDPC) codes, even for those that are rank deficient. The approach is based on constructing a set of codewords with the desired total rank by using minors of the parity-check matrix. We exemplify the method on several well-known and standard codes. Moreover, we explore the connections between the minors of the parity-check matrix and the known upper bound on minimum distance and provide a method to compute the rank of any parity-check matrix representing a QC-LDPC code, and hence the dimension of the code, by using the minors of the corresponding polynomial parity-check matrix.
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.
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.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.