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 International Symposium on Information Theory (ISIT)

Correcting combinations of errors and erasures with Euclidean geometry LDPC codes

Qiuju Diao, Ying Yu Tai, Shu Lin, Khaled Abdel-Ghaffar

Published Jul 1, 2013• 1 citation
Publisher

Abstract

It is shown that Euclidean geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.

DOI
10.1109/isit.2013.6620514

Related papers

LDPC Codes on Partial Geometries: Construction, Trapping Set Structure, and Puncturing

Qiuju Diao, Ying Yu Tai, Shu Lin +1 more

Many known constructions of LDPC codes can be placed in a general framework using the notion of partial geometries. Based on this notion, the structure of such LDPC codes can be analyzed using a geometric approach that illuminates important properties of their parity-check matrices. In this approach, trapping sets are represented by subgeometries of the geometry used to construct the code. Based on the incidence relations between lines and points in this geometry, the structure of trapping sets is investigated. On the other hand, it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity-check matrix of an LDPC code. This relates trapping sets, represented by subgeometries, and punctured matrices, represented by the residual geometries. The null spaces of these punctured matrices are LDPC codes which inherit many of the good structural properties of the original code. Hence, new LDPC codes, with various lengths and rates, can be obtained by puncturing an LDPC code constructed based on a partial geometry. Furthermore, these punctured matrices and codes can be used in a two-phase decoding scheme to correct combinations of errors and erasures.

IEEE Transactions on Information Theory•Nov 19, 2013•41 citations

Separating erasures from errors for decoding

Khaled Abdel-Ghaffar, Jos H. Weber

Most decoding algorithms of linear codes, in general, are designed to correct or detect errors. However, many channels cause erasures in addition to errors. In principle, decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code. For any given linear code and any given maximum number of correctable erasures, we introduce parity-check matrices yielding parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. This allows for the separation of erasures from errors to facilitate decoding. The parity-check matrices typically have redundant rows. We give several constructions of such matrices and prove general bounds on their minimum sizes.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•15 citations

Burst-Correction Decoding of Cyclic LDPC Codes

Shumei Song, Shu Lin, Khaled Abdel-Ghaffar

LDPC codes have excellent performance when iteratively decoded based on sparse parity-check matrices over both the AWGN channel and the erasure channel. In this paper, we propose a simple burst-correction decoding scheme for cyclic LDPC codes based on the same sparse circulant parity-check matrices used to perform iterative decoding. Although the proposed scheme may not achieve the full burst-correcting capability of the codes, it is rather simple and fast. We also show that the full burst-correcting capabilities of some codes constructed from Euclidean and projective geometries are very close to the Reiger upper bound

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•4 citations

Construction of quasi-cyclic LDPC codes based on euclidean geometries

Yuanhua Liu, Meiling Zhang

A new method for constructing quasi-cyclic (QC) low-density parity-check (LDPC) codes based on the structural properties of Euclidean geometries is presented. The proposed method changes the incidence vectors of some lines in the original Euclidean geometry method to construct much more parity check matrices and results in a large class of QC-LDPC codes with girth of at least 6. Simulations show that some codes in this class have better performance and lower error floor than the original Euclidean geometry codes.

International Conference on Communications and Networking in China (ChinaCom)•Aug 1, 2013•1 citation

Construction of Structured LDPC Codes for AWGN and Binary Erasure Channels

Y.Y. Tai, Longwen Lan, Lingqi Zeng +3 more

An algebraic method for constructing structured regular and irregular LDPC codes with girth at least six is presented. The parity check matrices of the proposed codes are arrays of permutation matrices obtained by decomposing Euclidean geometries. Masking is applied to the resultant parity check matrices to yield codes with various degree distributions. The proposed codes perform well over AWGN channels and binary erasure channels when decoded iteratively.

•Dec 10, 2005•2 citations

Small stopping sets in projective low-density parity-check codes

Yuichiro Fujiwara, Yu Tsunoda

It is known that redundant parity-check equations can improve the performance of an LDPC code by reducing the number of harmful substructures in the parity-check matrix. However, it is a difficult problem to design a parity-check matrix in such a way that it avoids substructures that are known to be harmful to iterative decoding while keeping the number of redundant parity-check equations moderate and ensuring other desirable properties. We explicitly give redundant parity-check matrices for cyclic regular LDPC codes of length n and minimum distance d ~ √n in which there are only n parity-check equations but no stopping sets of size d+1 or smaller except for those that correspond to the nonzero codewords of the smallest weight. We do this by showing that the well-known projective LDPC codes from the incidence matrices of projective planes PG(2, q) with q even have this property. This result may give insight into how the small number of redundant parity-check equations in the geometric LDPC codes may be contributing to the good performance reported in the literature. We also give a slightly improved upper bound on the size of a smallest generic erasure correcting set.

IEEE Information Theory Workshop (ITW)•Aug 1, 2019•0 citations
It is shown that Euclidean geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.
Many known constructions of LDPC codes can be placed in a general framework using the notion of partial geometries. Based on this notion, the structure of such LDPC codes can be analyzed using a geometric approach that illuminates important properties of their parity-check matrices. In this approach, trapping sets are represented by subgeometries of the geometry used to construct the code. Based on the incidence relations between lines and points in this geometry, the structure of trapping sets is investigated. On the other hand, it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity-check matrix of an LDPC code. This relates trapping sets, represented by subgeometries, and punctured matrices, represented by the residual geometries. The null spaces of these punctured matrices are LDPC codes which inherit many of the good structural properties of the original code. Hence, new LDPC codes, with various lengths and rates, can be obtained by puncturing an LDPC code constructed based on a partial geometry. Furthermore, these punctured matrices and codes can be used in a two-phase decoding scheme to correct combinations of errors and erasures.
Most decoding algorithms of linear codes, in general, are designed to correct or detect errors. However, many channels cause erasures in addition to errors. In principle, decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code. For any given linear code and any given maximum number of correctable erasures, we introduce parity-check matrices yielding parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. This allows for the separation of erasures from errors to facilitate decoding. The parity-check matrices typically have redundant rows. We give several constructions of such matrices and prove general bounds on their minimum sizes.
LDPC codes have excellent performance when iteratively decoded based on sparse parity-check matrices over both the AWGN channel and the erasure channel. In this paper, we propose a simple burst-correction decoding scheme for cyclic LDPC codes based on the same sparse circulant parity-check matrices used to perform iterative decoding. Although the proposed scheme may not achieve the full burst-correcting capability of the codes, it is rather simple and fast. We also show that the full burst-correcting capabilities of some codes constructed from Euclidean and projective geometries are very close to the Reiger upper bound
A new method for constructing quasi-cyclic (QC) low-density parity-check (LDPC) codes based on the structural properties of Euclidean geometries is presented. The proposed method changes the incidence vectors of some lines in the original Euclidean geometry method to construct much more parity check matrices and results in a large class of QC-LDPC codes with girth of at least 6. Simulations show that some codes in this class have better performance and lower error floor than the original Euclidean geometry codes.
It is known that redundant parity-check equations can improve the performance of an LDPC code by reducing the number of harmful substructures in the parity-check matrix. However, it is a difficult problem to design a parity-check matrix in such a way that it avoids substructures that are known to be harmful to iterative decoding while keeping the number of redundant parity-check equations moderate and ensuring other desirable properties. We explicitly give redundant parity-check matrices for cyclic regular LDPC codes of length n and minimum distance d ~ √n in which there are only n parity-check equations but no stopping sets of size d+1 or smaller except for those that correspond to the nonzero codewords of the smallest weight. We do this by showing that the well-known projective LDPC codes from the incidence matrices of projective planes PG(2, q) with q even have this property. This result may give insight into how the small number of redundant parity-check equations in the geometric LDPC codes may be contributing to the good performance reported in the literature. We also give a slightly improved upper bound on the size of a smallest generic erasure correcting set.