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

Journal of Algebraic Systems

A REDUCTION IN THE SEARCH SPACE OF QC-LDPC CODES WITH GIRTH 8

Farzane Amirzade, Meysam Alishahi, Mohammad‐Reza Sadeghi

Published Jan 1, 2020• 0 citations
Publisher

Abstract

In this paper, we define a structure to obtain exponent matrices of girth-8 QC-LDPC codes with column weight 3. Using the difference matrices introduced by Amirzade et al., we investigate necessary and sufficient conditions which result in a Tanner graph with girth 8. Our proposed method contributes to reduce the search space in recognizing the elements of an exponent matrix. In fact, in this method we only search to obtain one row of an exponent matrix. The other rows are multiplications of that row.

DOI
10.22044/jas.2019.8086.1397

Related papers

Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices

Farzane Amirzade, Mohammad‐Reza Sadeghi

In this paper, we define two “difference matrices” which correspond to an exponent matrix. We present necessary and sufficient conditions for these difference matrices to have quasi-cyclic low-density parity-check codes (QC-LDPC) codes with a certain girth. We achieve all non-isomorphic QC-LDPC codes with the shortest length, girth 6, the column weight, m = 4, and the row weight, 5 ≤ n ≤ 11. Moreover, a method to obtain an exponent matrix with girth 10 is presented which reduces the complexity of the search algorithm. If the first row and the first column of the exponent matrix are all-zero, then by applying our method we do not need to test Fossorier's lemma to avoid 6-cycles and 8-cycles. For girth 10, we also provide a lower bound on the lifting degree which is tighter than the existing bound. For girth 12, a new lower bound on the lifting degree is achieved.

IEEE Access•Jan 1, 2018•28 citations

Efficient Search of QC-LDPC Codes with Girths 6 and 8 and Free of Elementary Trapping Sets with Small Size

Farzane Amirzade, Mohammad‐Reza Sadeghi

One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An $(a,b)$ elementary trapping set, or simply an ETS where $a$ is the size and $b$ is the number of degree-one check nodes and $\frac{b}{a}<1$, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected $(3,n)$-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.

arXiv:1803.08141•Mar 21, 2018•4 citations

Several Explicit Constructions for (3,L) QC-LDPC Codes with Girth at Least Eight

Guohua Zhang, Rong Xia Sun, Xinmei Wang

Some explicit constructions for exponent matrices which correspond to Tanner graphs with girth at least eight are proposed. From these results, for any row weight of L>3, (3,L) quasi-cyclic LDPC codes with girth at least eight are constructed for any circulant permutation matrix (CPM) size P>L(L+mod(L,2))/2.

IEEE Communications Letters•Jul 16, 2013•29 citations

Necessary and Sufficient Girth Conditions for LDPC Tanner Graphs with Denser Protographs

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

This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix obtained from it.

arXiv:2106.00861•International Symposium on Turbo Codes and Iterative Information Processing•Aug 30, 2021•2 citations

QC-LDPC Codes With Large Column Weight and Free of Small Size ETSs

Farzane Amirzade, Mohammad‐Reza Sadeghi, Daniel Panario

An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we call$\textit {rainbow 8-cycle}$causes the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.

IEEE Communications Letters•Dec 27, 2021•11 citations

Quasi-Cyclic Protograph-Based Raptor-Like LDPC Codes With Girth 6 and Shortest Length

Farzane Amirzade, Mohammad‐Reza Sadeghi, Daniel Panario

We consider multiple-edge QC-LDPC codes with a base matrix of large size. We propose a new method, the degree reduction method, to obtain exponent matrices of these codes which considerably reduces the complexity of the search algorithm. We also provide a necessary and sufficient condition to avoid 4-cycles from occurrence in the Tanner graph of codes obtained using our method. Then, we apply our method to quasi-cyclic protograph-based Raptor-Like LDPC (QC-PBRL-LDPC) codes whose base matrices are multiple-edge. Numerical results show that as a consequence of this study we can obtain the minimum lifting degree of QC-PBRL-LDPC codes with girth at least 6. Thus, the lengths of the obtained codes are much smaller than those of their counterpart short-length codes in the literature.

IEEE International Symposium on Information Theory (ISIT)•Jul 12, 2021•6 citations
In this paper, we define a structure to obtain exponent matrices of girth-8 QC-LDPC codes with column weight 3. Using the difference matrices introduced by Amirzade et al., we investigate necessary and sufficient conditions which result in a Tanner graph with girth 8. Our proposed method contributes to reduce the search space in recognizing the elements of an exponent matrix. In fact, in this method we only search to obtain one row of an exponent matrix. The other rows are multiplications of that row.
In this paper, we define two “difference matrices” which correspond to an exponent matrix. We present necessary and sufficient conditions for these difference matrices to have quasi-cyclic low-density parity-check codes (QC-LDPC) codes with a certain girth. We achieve all non-isomorphic QC-LDPC codes with the shortest length, girth 6, the column weight, m = 4, and the row weight, 5 ≤ n ≤ 11. Moreover, a method to obtain an exponent matrix with girth 10 is presented which reduces the complexity of the search algorithm. If the first row and the first column of the exponent matrix are all-zero, then by applying our method we do not need to test Fossorier's lemma to avoid 6-cycles and 8-cycles. For girth 10, we also provide a lower bound on the lifting degree which is tighter than the existing bound. For girth 12, a new lower bound on the lifting degree is achieved.
One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An (a,b)(a,b)(a,b) elementary trapping set, or simply an ETS where aaa is the size and bbb is the number of degree-one check nodes and ba<1\frac{b}{a}<1ab​<1, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected (3,n)(3,n)(3,n)-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.
This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix obtained from it.
An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we callrainbow 8-cycle\textit {rainbow 8-cycle}rainbow 8-cyclecauses the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.
We consider multiple-edge QC-LDPC codes with a base matrix of large size. We propose a new method, the degree reduction method, to obtain exponent matrices of these codes which considerably reduces the complexity of the search algorithm. We also provide a necessary and sufficient condition to avoid 4-cycles from occurrence in the Tanner graph of codes obtained using our method. Then, we apply our method to quasi-cyclic protograph-based Raptor-Like LDPC (QC-PBRL-LDPC) codes whose base matrices are multiple-edge. Numerical results show that as a consequence of this study we can obtain the minimum lifting degree of QC-PBRL-LDPC codes with girth at least 6. Thus, the lengths of the obtained codes are much smaller than those of their counterpart short-length codes in the literature.