Edge-Based Anisotropic Decoding for Generalized Bicycle Codes

Bane Vasi\'c; Boulat A. Bash; Dimitris Chytas; Paul N. Fessatidis

arxiv: 2605.03218 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cs.IT· math.IT

Edge-Based Anisotropic Decoding for Generalized Bicycle Codes

Dimitris Chytas , Paul N. Fessatidis , Boulat A. Bash , Bane Vasi\'c This is my paper

Pith reviewed 2026-05-08 17:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords generalized bicycle codesQLDPC codesanisotropic decodingmin-sum decodingdegeneracyautomorphismsedge-coloringTanner graph

0 comments

The pith

Edge-coloring eliminates all automorphisms in low-weight subgraphs of generalized bicycle codes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Generalized bicycle codes fail to reach their designed distance under iterative decoding because certain degenerate error patterns stay related by automorphisms that the decoder leaves unchanged. The paper characterizes this degeneracy through the structure of the Tanner graph and shows that assigning distinct colors to edges breaks every such automorphism in the low-weight stabilizer-induced subgraphs. The coloring is implemented by running min-sum decoding with color-dependent update rules, called edge-anisotropic decoding. Experiments on several GB codes confirm that this version corrects more errors than both the standard isotropic decoder and a coarser block-anisotropic variant, and the gains appear within a small number of iterations.

Core claim

For GB codes, edge-coloring eliminates all automorphisms in low-weight stabilizer-induced subgraphs. This enables edge-anisotropic min-sum decoding to achieve improved performance over isotropic and block-anisotropic decoding for several GB codes in a small number of iterations.

What carries the argument

Edge-coloring of the Tanner graph, implemented as edge-anisotropic min-sum decoding that applies distinct message-update rules according to edge color to destroy preserved automorphisms.

If this is right

Edge-coloring removes every automorphism from low-weight stabilizer-induced subgraphs in GB codes.
Edge-anisotropic min-sum decoding outperforms both isotropic and block-anisotropic decoding on tested GB codes.
The performance advantage appears within a small number of decoding iterations.
The graph-theoretic view ties persistent degeneracy directly to decoder-preserved automorphisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same edge-coloring construction may be worth testing on other QLDPC families whose Tanner graphs contain similar low-weight subgraphs.
Edge anisotropy could be combined with existing ordered-statistics or post-processing steps to further reduce the gap to maximum-likelihood performance.
If the symmetry-breaking assumption holds generally, the method supplies a low-overhead route to make sparse quantum codes practical at moderate block lengths.

Load-bearing premise

Harmful degenerate errors are precisely the ones related by automorphisms the decoder preserves, and edge-coloring can be added without introducing new failure modes or excessive complexity.

What would settle it

A GB code instance and error pattern where the edge-anisotropic decoder produces a higher frame-error rate than the isotropic decoder, or where an automorphism survives edge-coloring in a low-weight stabilizer subgraph.

Figures

Figures reproduced from arXiv: 2605.03218 by Bane Vasi\'c, Boulat A. Bash, Dimitris Chytas, Paul N. Fessatidis.

**Figure 1.** Figure 1: Induced subgraphs of stabilizers formed by linear combinations of view at source ↗

**Figure 2.** Figure 2: Logical error rate under X errors for isotropic MS, block-anisotropic MS, and an ensemble realization of edge-anisotropic MS on three representative GB codes: (a) the GB code of [18], (b) the [[288, 12, 18]] BB code, and (c) the A5 code [7]. Results are shown for iteration budgets I ∈ {10, 20, 50}. message stalling, as we observed degraded performance for values below 0.5. For all experiments, we fix the … view at source ↗

read the original abstract

Quantum low-density parity-check (QLDPC) codes provide non vanishing rates, distance scaling with the blocklength of the code, and facilitate fast iterative decoding because of their sparsity. However, in practice iterative decoding fails to exploit the distance of the code, because it cannot resolve the symmetries imposed by degeneracy. In this work, we provide a graph theoretic characterization of degeneracy for the family of generalized bicycle (GB) codes. This viewpoint shows that harmful degenerate error patterns persist whenever they remain related by automorphisms preserved by the decoder. Motivated by symmetry breaking via graph coloring, we compare three coloring approaches: no coloring, block-coloring, and edge-coloring. For GB codes, we show that edge-coloring can eliminate all automorphisms in low-weight stabilizer-induced subgraphs. We practically realize the coloring schemes as isotropic, block- anisotropic and edge-anisotropic min-sum (MS) decoding. Experimental results show that edge anisotropic min-sum decoding obtains improved performance over isotropic and block anisotropic decoding for several GB codes in a small number of iterations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Edge-coloring removes automorphisms from low-weight subgraphs in GB codes and gives a measurable lift to min-sum decoding in few iterations.

read the letter

The paper's core contribution is a graph-theoretic account of degeneracy in generalized bicycle codes: harmful error patterns survive iterative decoding when they are related by automorphisms that the decoder itself preserves. They show that edge-coloring the Tanner graph eliminates all such automorphisms in the low-weight stabilizer-induced subgraphs, then turn the idea into an edge-anisotropic min-sum decoder. Simulations on several GB codes indicate this version outperforms both the isotropic baseline and a block-anisotropic variant when the iteration budget is small. That is the concrete, usable result. The proof that edge-coloring clears the automorphisms is the part that feels new relative to prior symmetry-breaking work on QLDPC codes. The implementation is straightforward enough that it could be tried on other sparse codes without major re-engineering. The experimental section is limited to small iteration counts, which is reasonable given the target use case of fast decoding, but it leaves open how much the advantage persists once the decoder is allowed to run longer. The work stays within GB codes, so the scope is narrow, yet those codes remain one of the more practical QLDPC families. No internal contradictions or unstated assumptions that would collapse the argument appear in the structure. The citation pattern is appropriate and the claims rest on explicit graph properties plus simulation rather than fitted parameters. This is the kind of incremental but well-grounded piece that people building actual decoders for fault-tolerant architectures would want to see. I would bring it to a reading group focused on quantum LDPC constructions and send it out for peer review; the central claim is supported by both the proof and the reported experiments, even if later work will need to test broader iteration regimes and other code families.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a graph-theoretic characterization of degeneracy for generalized bicycle (GB) codes, arguing that harmful degenerate error patterns persist when they remain related by automorphisms preserved by the decoder. It compares three symmetry-breaking approaches (no coloring, block-coloring, edge-coloring) and proves that edge-coloring eliminates all automorphisms in low-weight stabilizer-induced subgraphs for GB codes. These schemes are realized as isotropic, block-anisotropic, and edge-anisotropic min-sum decoding, with simulations showing that the edge-anisotropic variant improves performance over the others for several GB codes, especially within a small number of iterations.

Significance. If the central claims hold, the work is significant for practical QLDPC decoding: the graph-coloring perspective supplies a concrete tool for analyzing and mitigating degeneracy, a known obstacle to achieving the full distance of sparse quantum codes. The explicit construction of edge-anisotropic min-sum decoding offers a low-overhead symmetry-breaking method that demonstrably accelerates convergence. The graph-theoretic proof that edge-coloring removes all relevant automorphisms is a clear technical contribution that could generalize to other code families.

minor comments (2)

The abstract asserts improved performance but does not quantify the gains (e.g., logical error-rate reduction at a given physical error rate or iteration count). A brief summary sentence with one or two representative numbers would help readers assess the practical impact before reading the full experimental section.
Notation for the three decoding variants (isotropic, block-anisotropic, edge-anisotropic) is introduced in the abstract; ensure the first use in the main text is accompanied by a short definition or pointer to the corresponding algorithm box.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We are pleased that the graph-theoretic characterization of degeneracy in generalized bicycle codes, the symmetry-breaking role of edge-coloring, and the performance gains of edge-anisotropic min-sum decoding are recognized as significant for practical QLDPC decoding.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives a graph-theoretic view of degeneracy for GB codes, proves that edge-coloring removes automorphisms from low-weight subgraphs, and implements this as edge-anisotropic min-sum decoding whose performance is then measured empirically. None of these steps reduce by construction to fitted inputs, self-referential definitions, or load-bearing self-citations; the coloring-to-decoder mapping is an explicit construction, and the reported gains are simulation outcomes rather than tautologies. The argument remains self-contained against external graph-coloring and decoding benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard graph theory applied to the Tanner graph of GB codes and on the assumption that min-sum decoding preserves certain automorphisms; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Degeneracy in GB codes can be characterized by automorphisms of the stabilizer-induced subgraphs that are preserved by the decoder.
Invoked to motivate symmetry breaking via coloring.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (washburn_uniqueness_aczel) J(x) = ½(x+x⁻¹)−1 uniqueness unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We generate a random pool of 100 candidate decoders and evaluate it on the harmful degenerate error patterns... by selecting five decoders that collectively give the best coverage within 10 iterations.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

Asymptotically good quantum and locally testable classical LDPC codes,

P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pp. 375–388

work page 2022
[2]

Quantum tanner codes,

A. Leverrier and G. Z ´emor, “Quantum tanner codes,” in2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2022, pp. 872–883

work page 2022
[3]

Preskill

J. Preskill, “Beyond nisq: The megaquop machine,”ACM Transactions on Quantum Computing, vol. 6, no. 3, p. 1–7, Apr. 2025. [Online]. Available: http://dx.doi.org/10.1145/3723153

work page doi:10.1145/3723153 2025
[4]

How to choose a decoder for a fault-tolerant quantum computer? the speed vs accuracy trade-off

N. Delfosse, A. Paz, A. Vaschillo, and K. M. Svore, “How to choose a decoder for a fault-tolerant quantum computer? the speed vs accuracy trade-off,” 2023. [Online]. Available: https://arxiv.org/abs/2310.15313

work page arXiv 2023
[5]

Trapping sets of quantum LDPC codes,

N. Raveendran and B. Vasi ´c, “Trapping sets of quantum LDPC codes,” Quantum, vol. 5, p. 562, Oct. 2021

work page 2021
[6]

Degeneracy and its impact on the decoding of sparse quantum codes,

P. Fuentes, J. Etxezarreta, P. Crespo, and J. Garcia-Frias, “Degeneracy and its impact on the decoding of sparse quantum codes,”IEEE Access, vol. 9, pp. 89 093–89 119, Jun. 2021

work page 2021
[7]

Degenerate quantum LDPC codes with good finite length performance,

P. Panteleev and G. Kalachev, “Degenerate quantum LDPC codes with good finite length performance,”Quantum, vol. 5, p. 585, Nov. 2021

work page 2021
[8]

arXiv preprint arXiv:2406.14527 (2024)

S. Wolanski and B. Barber, “Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes,” 2024. [Online]. Available: https://arxiv.org/abs/2406.14527

work page arXiv 2024
[9]

Quintavalle, Jens Eisert, Robert Wille, and Joschka Roffe

T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, “Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes,” 2024. [Online]. Available: https://arxiv.org/abs/2406.18655

work page arXiv 2024
[10]

Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation,

H. Yao, W. A. Laban, C. H ¨ager, A. G. i Amat, and H. D. Pfister, “Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation,”arXiv preprint arXiv:2308.13377, 2023

work page arXiv 2023
[11]

arXiv preprint arXiv:2403.18901 (2024)

A. Gong, S. Cammerer, and J. M. Renes, “Toward Low-latency Iterative Decoding of QLDPC Codes Under Circuit-Level Noise,” 2024. [Online]. Available: https://arxiv.org/abs/2403.18901

work page arXiv 2024
[12]

Restart belief: A general quantum ldpc decoder,

L. Valentini, D. Forlivesi, A. Talarico, and M. Chiani, “Restart belief: A general quantum ldpc decoder,” 2025. [Online]. Available: https://arxiv.org/abs/2511.13281

work page arXiv 2025
[13]

arXiv preprint arXiv:2506.01779 , year=

T. M ¨uller, T. Alexander, M. E. Beverland, M. B ¨uhler, B. R. Johnson, T. Maurer, and D. Vandeth, “Improved belief propagation is sufficient for real-time decoding of quantum memory,” 2025. [Online]. Available: https://arxiv.org/abs/2506.01779

work page arXiv 2025
[14]

Enhanced Min- Sum Decoding of Quantum Codes Us- ing Previous Iteration Dynamics

D. Chytas, N. Raveendran, and B. Vasic, “Enhanced min-sum decoding of quantum codes using previous iteration dynamics,” 2025. [Online]. Available: https://arxiv.org/abs/2501.05021

work page arXiv 2025
[15]

Enhanced min-sum decoding of quantum codes with iteration dynamics memory,

D. Chytas, N. Raveendran, and B. Vasi ´c, “Enhanced min-sum decoding of quantum codes with iteration dynamics memory,” in2025 IEEE International Symposium on Information Theory (ISIT), 2025, pp. 1– 6

work page 2025
[16]

Linear time iterative decoders for hypergraph-product and lifted-product codes,

A. K. Pradhan, N. Raveendran, N. Rengaswamy, and B. Vasi ´c, “Linear time iterative decoders for hypergraph-product and lifted-product codes,” 2025. [Online]. Available: https://arxiv.org/abs/2504.01728

work page arXiv 2025
[17]

High-threshold and low-overhead fault-tolerant quantum memory,

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,”Nature, vol. 627, no. 8005, pp. 778–782, 2024

work page 2024
[18]

Wang and L

R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes,” 2022. [Online]. Available: https://arxiv.org/abs/2203.17216

work page arXiv 2022
[19]

Good quantum error-correcting codes exist,

A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Physical Review A, vol. 54, no. 2, p. 1098, 1996

work page 1996
[20]

Stabilizer codes and quantum error correction,

D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D. dissertation, California Institute of Technology, 1997

work page 1997
[21]

Sparse-graph codes for quantum error correction,

D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-graph codes for quantum error correction,”IEEE Transactions on Information Theory, vol. 50, no. 10, pp. 2315–2330, Oct. 2004

work page 2004
[22]

Iterative decoding beyond belief propagation,

S. K. Planjery, S. K. Chilappagari, B. Vasi ´c, D. Declercq, and L. Dan- jean, “Iterative decoding beyond belief propagation,” in2010 Informa- tion Theory and Applications Workshop (ITA), 2010, pp. 1–10

work page 2010
[23]

Practical graph isomorphism, ii,

B. D. McKay and A. Piperno, “Practical graph isomorphism, ii,” 2013. [Online]. Available: https://arxiv.org/abs/1301.1493

work page arXiv 2013
[24]

Reduced-complexity decoding of LDPC codes,

J. Chen, A. Dholakia, E. Eleftheriou, M. Fossorier, and X.-Y . Hu, “Reduced-complexity decoding of LDPC codes,”IEEE Trans. Commun., vol. 53, no. 8, pp. 1288–1299, Aug. 2005

work page 2005

[1] [1]

Asymptotically good quantum and locally testable classical LDPC codes,

P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pp. 375–388

work page 2022

[2] [2]

Quantum tanner codes,

A. Leverrier and G. Z ´emor, “Quantum tanner codes,” in2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2022, pp. 872–883

work page 2022

[3] [3]

Preskill

J. Preskill, “Beyond nisq: The megaquop machine,”ACM Transactions on Quantum Computing, vol. 6, no. 3, p. 1–7, Apr. 2025. [Online]. Available: http://dx.doi.org/10.1145/3723153

work page doi:10.1145/3723153 2025

[4] [4]

How to choose a decoder for a fault-tolerant quantum computer? the speed vs accuracy trade-off

N. Delfosse, A. Paz, A. Vaschillo, and K. M. Svore, “How to choose a decoder for a fault-tolerant quantum computer? the speed vs accuracy trade-off,” 2023. [Online]. Available: https://arxiv.org/abs/2310.15313

work page arXiv 2023

[5] [5]

Trapping sets of quantum LDPC codes,

N. Raveendran and B. Vasi ´c, “Trapping sets of quantum LDPC codes,” Quantum, vol. 5, p. 562, Oct. 2021

work page 2021

[6] [6]

Degeneracy and its impact on the decoding of sparse quantum codes,

P. Fuentes, J. Etxezarreta, P. Crespo, and J. Garcia-Frias, “Degeneracy and its impact on the decoding of sparse quantum codes,”IEEE Access, vol. 9, pp. 89 093–89 119, Jun. 2021

work page 2021

[7] [7]

Degenerate quantum LDPC codes with good finite length performance,

P. Panteleev and G. Kalachev, “Degenerate quantum LDPC codes with good finite length performance,”Quantum, vol. 5, p. 585, Nov. 2021

work page 2021

[8] [8]

arXiv preprint arXiv:2406.14527 (2024)

S. Wolanski and B. Barber, “Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes,” 2024. [Online]. Available: https://arxiv.org/abs/2406.14527

work page arXiv 2024

[9] [9]

Quintavalle, Jens Eisert, Robert Wille, and Joschka Roffe

T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, “Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes,” 2024. [Online]. Available: https://arxiv.org/abs/2406.18655

work page arXiv 2024

[10] [10]

Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation,

H. Yao, W. A. Laban, C. H ¨ager, A. G. i Amat, and H. D. Pfister, “Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation,”arXiv preprint arXiv:2308.13377, 2023

work page arXiv 2023

[11] [11]

arXiv preprint arXiv:2403.18901 (2024)

A. Gong, S. Cammerer, and J. M. Renes, “Toward Low-latency Iterative Decoding of QLDPC Codes Under Circuit-Level Noise,” 2024. [Online]. Available: https://arxiv.org/abs/2403.18901

work page arXiv 2024

[12] [12]

Restart belief: A general quantum ldpc decoder,

L. Valentini, D. Forlivesi, A. Talarico, and M. Chiani, “Restart belief: A general quantum ldpc decoder,” 2025. [Online]. Available: https://arxiv.org/abs/2511.13281

work page arXiv 2025

[13] [13]

arXiv preprint arXiv:2506.01779 , year=

T. M ¨uller, T. Alexander, M. E. Beverland, M. B ¨uhler, B. R. Johnson, T. Maurer, and D. Vandeth, “Improved belief propagation is sufficient for real-time decoding of quantum memory,” 2025. [Online]. Available: https://arxiv.org/abs/2506.01779

work page arXiv 2025

[14] [14]

Enhanced Min- Sum Decoding of Quantum Codes Us- ing Previous Iteration Dynamics

D. Chytas, N. Raveendran, and B. Vasic, “Enhanced min-sum decoding of quantum codes using previous iteration dynamics,” 2025. [Online]. Available: https://arxiv.org/abs/2501.05021

work page arXiv 2025

[15] [15]

Enhanced min-sum decoding of quantum codes with iteration dynamics memory,

D. Chytas, N. Raveendran, and B. Vasi ´c, “Enhanced min-sum decoding of quantum codes with iteration dynamics memory,” in2025 IEEE International Symposium on Information Theory (ISIT), 2025, pp. 1– 6

work page 2025

[16] [16]

Linear time iterative decoders for hypergraph-product and lifted-product codes,

A. K. Pradhan, N. Raveendran, N. Rengaswamy, and B. Vasi ´c, “Linear time iterative decoders for hypergraph-product and lifted-product codes,” 2025. [Online]. Available: https://arxiv.org/abs/2504.01728

work page arXiv 2025

[17] [17]

High-threshold and low-overhead fault-tolerant quantum memory,

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,”Nature, vol. 627, no. 8005, pp. 778–782, 2024

work page 2024

[18] [18]

Wang and L

R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes,” 2022. [Online]. Available: https://arxiv.org/abs/2203.17216

work page arXiv 2022

[19] [19]

Good quantum error-correcting codes exist,

A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Physical Review A, vol. 54, no. 2, p. 1098, 1996

work page 1996

[20] [20]

Stabilizer codes and quantum error correction,

D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D. dissertation, California Institute of Technology, 1997

work page 1997

[21] [21]

Sparse-graph codes for quantum error correction,

D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-graph codes for quantum error correction,”IEEE Transactions on Information Theory, vol. 50, no. 10, pp. 2315–2330, Oct. 2004

work page 2004

[22] [22]

Iterative decoding beyond belief propagation,

S. K. Planjery, S. K. Chilappagari, B. Vasi ´c, D. Declercq, and L. Dan- jean, “Iterative decoding beyond belief propagation,” in2010 Informa- tion Theory and Applications Workshop (ITA), 2010, pp. 1–10

work page 2010

[23] [23]

Practical graph isomorphism, ii,

B. D. McKay and A. Piperno, “Practical graph isomorphism, ii,” 2013. [Online]. Available: https://arxiv.org/abs/1301.1493

work page arXiv 2013

[24] [24]

Reduced-complexity decoding of LDPC codes,

J. Chen, A. Dholakia, E. Eleftheriou, M. Fossorier, and X.-Y . Hu, “Reduced-complexity decoding of LDPC codes,”IEEE Trans. Commun., vol. 53, no. 8, pp. 1288–1299, Aug. 2005

work page 2005