An efficient exhaustive low-weight codeword search for structured LDPC codes

Khatami, Seyed Mehrdad; Danjean, Ludovic; Nguyen, Dung Viet; Vasić, Bane

doi:10.1109/ita.2013.6502981

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…Enumeration of cycles and their combinations also allows to find harmful classical-type TSs in the QLDPC code. Unlike these classicaltype TSs, the search for symmetric stabilizer TSs requires a different approach of finding low-weight codeword sub-graphs [31] with additional symmetry constraints. In the case of CSS codes, the H Z even-weight stabilizer generators are examples of symmetric stabilizer TSs for iterative decoding over the Tanner graph of H X matrix and viceversa.…”

Section: Searching For Quantum Trapping Setsmentioning

confidence: 99%

Trapping Sets of Quantum LDPC Codes

Raveendran

Vasić

2021

Quantum

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Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.

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Section: Searching For Quantum Trapping Setsmentioning

confidence: 99%

Trapping Sets of Quantum LDPC Codes

Raveendran

Vasić

2021

Quantum

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“…Enumeration of cycles and their combinations also allows to find harmful classical-type TSs in the QLDPC code. Unlike these classical-type TSs, the search for symmetric stabilizer TSs requires a different approach of finding lowweight codeword sub-graphs [27] with additional symmetry constraints. In the case of CSS codes, the H Z even-weight stabilizer generators are examples of symmetric stabilizer TSs for iterative decoding over the Tanner graph of H X matrix and vice-versa.…”

Section: B Searching For Quantum Trapping Setsmentioning

confidence: 99%

Trapping Sets of Quantum LDPC Codes

Raveendran,

Vasić

2020

Preprint

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Iterative decoders for finite length quantum lowdensity parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance, and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. Conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC code and decoder. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.

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“…Furthermore, in [25], [75], [6], analytical upper bounds on the minimum distance of quasi cyclic (QC) LDPC codes were derived. In another direction, search algorithms have been proposed to find d min of LDPC codes, or to provide lower and/or upper bounds on d min [41], [68], [10], [49], [39], [50], [6]. Error impulse method, originally proposed for turbo codes, was used in [41], and later modified in [10], to estimate d min of LDPC codes.…”

Section: Related Workmentioning

confidence: 99%

“…As a result, in [49], only the minimum distances of LDPC codes with moderate rate (R = 0.5) and relatively short block length (n = 512) are reported. In [50], the authors studied the topological structure of codewords of weight a with a ≤ 14, in regular LDPC codes with variable degree 3 and girth 8, and proposed a search algorithm to find such low-weight codewords. The algorithm is, however, limited to only regular LDPC codes with variable degree 3 and girth 8.…”

Section: Related Workmentioning

confidence: 99%

Characterization of Problematic Graphical Structures of LDPC Codes and the Corresponding Efficient Search Algorithms

Toroghi¹

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In this thesis, we study all the problematic graphical structures which play important roles in the error floor performance and error correction capability of low-density parity-check (LDPC) codes. These graphical structures are: leafless elementary trapping sets (LETSs), elementary trapping sets (ETSs), non-elementary trapping sets (NETSs), stopping sets and codewords. Recently, Karimi and Banihashemi proposed a characterization of LETSs, which was based on viewing a LETS as a layered superset (LSS) of a short cycle in the code's Tanner graph. However this characterization was not exhaustive. In this work, first, we complement LSS characterization by demonstrating how the remaining structures of LETSs can be characterized. Then, we propose a new characterization for LETSs of variable-regular LDPC codes. Compared to the LSS-based characterization, which is based on a single LSS expansion technique, the new characterization involves two additional expansion techniques. The introduction of the new techniques mitigates the search efficiency problem that LSS-based characterization/search suffers from. We prove that using the three expansion techniques, any LETS structure can be obtained starting from a simple cycle, no matter how large the size of the structure a or the number of its unsatisfied check nodes b are, i.e., the characterization is exhaustive. We also demonstrate that for the proposed characterization/search the length of the short cycles required to be enumerated is minimal. We also prove that the three expansion techniques, proposed here, are the only expansions needed for characterization of LETS structures starting from simple cycles in the graph.Moreover, we generalize the proposed approach of variable-regular to irregular LDPC codes. We explain how the characterization of LETS structures in variableregular graphs can be used to characterize the LETS structures of irregular graphs (in a given range of a and b values, exhaustively). This characterization corresponds to an exhaustive search algorithm. However, this approach is not applicable to irregular LDPC codes, if we are interested to find LETSs in a relatively wide range of iii First, I would like to thank my supervisor, Professor Amir H. Banihashemi for leading and encouraging me during my research. His guidance helped me in all the time of research and writing of this thesis.Thanks must also go to my committee members for their helpful suggestions and discussions.

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An efficient exhaustive low-weight codeword search for structured LDPC codes

Cited by 7 publications

References 31 publications

Trapping Sets of Quantum LDPC Codes

Trapping Sets of Quantum LDPC Codes

Trapping Sets of Quantum LDPC Codes

Characterization of Problematic Graphical Structures of LDPC Codes and the Corresponding Efficient Search Algorithms

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