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)

Separating erasures from errors for decoding

Khaled Abdel-Ghaffar, Jos H. Weber

Published Jul 1, 2008• 15 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2008.4594979

Related papers

An upper bound on the separating redundancy of linear block codes

Khaled Abdel-Ghaffar, Jos H. Weber

Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating parity-check matrices. For a given maximum number of correctable erasures, such matrices yield 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. Separating parity-check matrices typically have redundant rows. An upper bound on the minimum number of rows in separating parity-check matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2010•2 citations

Correcting combinations of errors and erasures with Euclidean geometry LDPC codes

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

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.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2013•1 citation

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

Thorsten Hehn, Olgica Milenković, Stefan Laendner +1 more

We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.

arXiv:cs/0702050•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•3 citations

A decoding algorithm for LDPC codes over erasure channels with sporadic errors

Gianluigi Liva, Enrico Paolini, Balázs Matuz +1 more

An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multiple-erasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.

Allerton Conference on Communication, Control, and Computing•Sep 1, 2010•11 citations

On error correction with errors in both the channel and syndrome

Serge G. Vladut, Grigory A. Kabatiansky, V. V. Lomakov

We address the problem of error correction by linear block codes under the assumption that the syndrome of a received vector is found with errors. We propose a construction of parity-check matrices which allow to solve the syndrome equation even with an erroneous syndrome, in particular, parity-check matrices with minimum redundancy, which are analogs of Reed-Solomon codes for this problem. We also establish analogs of classical coding theory bounds, namely the Hamming, Singleton, and Gilbert-Varshamov bounds. We show that the new problem can be considered as a generalization of the well-known Ulam’s problem on searching with a lie and as a discrete analog of the compressed sensing problem.

Problems of Information Transmission•Apr 1, 2015•11 citations

Decoding of low-density codes with parity-check matrices composed of permutation matrices in an erasure channel

D. K. Zigangirov, Kamil Sh. Zigangirov

A lower bound for the number of iteratively correctable erasures is given, with application to the ensemble of LDPC codes with parity-check matrices composed of permutation matrices [1]. We assume that the Zyablov-Pinsker iterative decoding algorithm [2] is used. Its complexity is O ( N log N ), where N is the block length.

Problems of Information Transmission•Jun 1, 2006•4 citations
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.
Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating parity-check matrices. For a given maximum number of correctable erasures, such matrices yield 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. Separating parity-check matrices typically have redundant rows. An upper bound on the minimum number of rows in separating parity-check matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length.
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.
An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multiple-erasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.
We address the problem of error correction by linear block codes under the assumption that the syndrome of a received vector is found with errors. We propose a construction of parity-check matrices which allow to solve the syndrome equation even with an erroneous syndrome, in particular, parity-check matrices with minimum redundancy, which are analogs of Reed-Solomon codes for this problem. We also establish analogs of classical coding theory bounds, namely the Hamming, Singleton, and Gilbert-Varshamov bounds. We show that the new problem can be considered as a generalization of the well-known Ulam’s problem on searching with a lie and as a discrete analog of the compressed sensing problem.