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arxiv: 2605.01942 · v1 · submitted 2026-05-03 · 💻 cs.IT · math.IT· quant-ph

Combinatorial Analysis of Dyadic and Quasi-Dyadic Codes

Anthony G\'omez-Fonseca , Gretchen L. Matthews , Kirsten D. Morris , Tefjol Pllaha This is my paper

Pith reviewed 2026-05-08 19:29 UTC · model grok-4.3

classification 💻 cs.IT math.ITquant-ph
keywords dyadic matricesquasi-dyadic codesLDPC codesQLDPC codesTanner graphsshort cyclesabsorbing setsbelief propagation
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The pith

Dyadic matrices let researchers map and reduce short cycles in LDPC Tanner graphs by counting protograph walks.

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

The paper builds an algebraic framework around dyadic and quasi-dyadic matrices—compact 2^ℓ by 2^ℓ binary matrices given by a signature row and equipped with recursive block structure—to construct and analyze LDPC and QLDPC codes. It uses that structure to translate cycles appearing in lifted Tanner graphs into tailless backtrackless closed walks on a protograph, which yields fast ways to count and limit 4-, 6-, and 8-cycles. Construction routines based on forbidden shift sets either raise girth when possible or cut the number of shortest cycles otherwise, while also identifying absorbing-set configurations that hurt iterative decoding. Simulations then show that lowering cycle multiplicity alone produces clear belief-propagation gains even when girth stays fixed.

Core claim

The recursive block structure of dyadic matrices forms a commutative ring that lets cycles in any lifted Tanner graph be read off directly from tailless backtrackless closed walks in the underlying protograph; this correspondence supplies implementable algorithms that enumerate short cycles, support PEG-style constructions using forbidden shifts to minimize multiplicity at a target girth, characterize boundary absorbing sets that produce many (a,0) instances, and generate CSS codes satisfying commutation relations by construction, with belief-propagation tests confirming decoding improvement from multiplicity reduction alone.

What carries the argument

The direct correspondence between cycles in lifted Tanner graphs and tailless backtrackless closed walks in the protograph, made possible by the recursive block decomposition of dyadic matrices.

If this is right

  • Dyadic-aware PEG constructions can achieve lower short-cycle counts than girth-maximizing methods that ignore the algebraic structure.
  • Explicit enumeration identifies and avoids absorbing-set layouts that create many (a,0) instances in boundary cases.
  • CSS constructions built from dyadic signatures satisfy the required commutation relations automatically.
  • Belief-propagation decoding performance improves when short-cycle multiplicity drops, even if the overall girth remains unchanged.

Where Pith is reading between the lines

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

  • The same walk-enumeration technique could be adapted to other translation-invariant or recursively structured matrix families used in classical LDPC design.
  • Lower multiplicity at modest girth may allow larger QLDPC codes to be decoded reliably before error floors appear.
  • Forbidden-shift rules derived here might combine with existing algebraic constructions to produce hybrid codes with both good distance and controlled cycle profiles.

Load-bearing premise

The recursive block structure of dyadic matrices lets cycles in the lifted Tanner graph be read off as tailless backtrackless closed walks in the protograph.

What would settle it

A concrete dyadic code in which the short cycles predicted by enumerating protograph walks fail to match the actual cycles present in its lifted Tanner graph, or a family of such codes where belief-propagation error rates show no improvement after multiplicity reduction at fixed girth.

Figures

Figures reproduced from arXiv: 2605.01942 by Anthony G\'omez-Fonseca, Gretchen L. Matthews, Kirsten D. Morris, Tefjol Pllaha.

Figure 1
Figure 1. Figure 1: Subfigure (a): Two dyadic matrices with signature weight 4 and view at source ↗
read the original abstract

Quantum low-density parity-check (QLDPC) codes offer a promising route to scalable fault-tolerant quantum computation, but their performance under iterative decoding is strongly influenced by short-cycle structure and other harmful subgraphs in the associated Tanner graphs. This paper develops an algebraic framework for constructing and analyzing (Q)LDPC codes from dyadic and quasi-dyadic matrices-translation-invariant $2^\ell \times 2^\ell$ binary matrices specified compactly by a signature row and forming a commutative ring with recursive block structure. Leveraging this structure, we relate cycles in lifted Tanner graphs to tailless backtrackless closed walks in the protograph and derive efficient, implementable methods to enumerate and control short cycles (notably $4$-, $6$-, and $8$-cycles). We introduce dyadic-aware PEG-style construction algorithms that use forbidden sets of shifts to maximize attainable girth when possible and otherwise minimize the multiplicity of the shortest cycles at the target girth. Motivated by error-floor phenomena, we further characterize and explicitly enumerate absorbing sets in key dyadic layout boundary cases, identifying configurations that induce abundant $(a,0)$-absorbing sets. Finally, we propose CSS-oriented dyadic constructions that satisfy commutation constraints by design and demonstrate via belief-propagation simulations that reducing short-cycle multiplicity can yield substantial decoding gains even when girth cannot be increased.

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.

Referee Report

2 major / 2 minor

Summary. The paper develops an algebraic framework for constructing and analyzing (Q)LDPC codes from dyadic and quasi-dyadic matrices, which are translation-invariant 2^ℓ × 2^ℓ binary matrices specified by a signature row and forming a commutative ring with recursive block structure. It relates cycles in lifted Tanner graphs to tailless backtrackless closed walks in the protograph, derives efficient enumeration methods for short cycles (4-, 6-, and 8-cycles), introduces dyadic-aware PEG-style construction algorithms that maximize girth or minimize shortest-cycle multiplicity, explicitly enumerates absorbing sets in key boundary cases, proposes CSS-oriented constructions satisfying commutation constraints by design, and demonstrates via belief-propagation simulations that reducing short-cycle multiplicity yields decoding gains even when girth cannot be increased.

Significance. If the cycle-mapping derivations and enumeration algorithms hold, the work supplies a practical combinatorial toolkit for controlling harmful subgraphs in QLDPC Tanner graphs, which is directly relevant to improving error floors in quantum error correction. The use of dyadic commutativity for efficient walk enumeration, the explicit absorbing-set characterization, and the simulation evidence of gains at fixed girth are notable strengths that extend standard protograph techniques in a concrete, implementable direction.

major comments (2)
  1. [Simulations] Simulations section: the reported decoding gains from reduced cycle multiplicity rest on belief-propagation runs whose block length, number of iterations, Monte Carlo trial count, and exact baseline constructions (standard PEG, random dyadic, etc.) are not stated with sufficient precision to allow independent verification of the magnitude of the gains or to rule out confounding factors such as degree distribution differences.
  2. [Absorbing Sets] Absorbing-sets enumeration: the identification of configurations inducing abundant (a,0)-absorbing sets is restricted to specific dyadic layout boundary cases; it is unclear whether the same enumeration procedure extends without modification to the quasi-dyadic case or to the CSS constructions, which would be necessary to support the broader claim that cycle control alone suffices to mitigate error floors.
minor comments (2)
  1. [Introduction] The abstract and introduction use the term 'dyadic-aware PEG-style' without an immediate forward reference to the precise forbidden-shift rule or pseudocode; adding a short algorithmic box or numbered steps would improve readability.
  2. [Cycle Enumeration] Notation for the signature row and the recursive block decomposition is introduced compactly but the precise mapping from signature entries to the 2^ℓ × 2^ℓ matrix entries is not restated in the cycle-enumeration section; a one-line reminder would prevent reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Simulations] Simulations section: the reported decoding gains from reduced cycle multiplicity rest on belief-propagation runs whose block length, number of iterations, Monte Carlo trial count, and exact baseline constructions (standard PEG, random dyadic, etc.) are not stated with sufficient precision to allow independent verification of the magnitude of the gains or to rule out confounding factors such as degree distribution differences.

    Authors: We agree that the simulation details must be stated with greater precision to support independent verification. In the revised manuscript we will explicitly report the block lengths employed, the number of belief-propagation iterations, the Monte Carlo trial counts per SNR point, and the precise baseline constructions (including standard PEG and random dyadic matrices), while confirming that degree distributions are matched across compared ensembles. revision: yes

  2. Referee: [Absorbing Sets] Absorbing-sets enumeration: the identification of configurations inducing abundant (a,0)-absorbing sets is restricted to specific dyadic layout boundary cases; it is unclear whether the same enumeration procedure extends without modification to the quasi-dyadic case or to the CSS constructions, which would be necessary to support the broader claim that cycle control alone suffices to mitigate error floors.

    Authors: The explicit enumeration is presented for dyadic boundary cases because the recursive block structure permits a complete combinatorial characterization there. The underlying walk-enumeration technique extends directly to quasi-dyadic matrices via the same signature-row representation. For the CSS constructions the commutation constraints are satisfied by design, so the absorbing-set analysis on the component dyadic protographs remains applicable. We will add a clarifying paragraph in the revised version that states these extensions and the scope of the broader claim. revision: partial

Circularity Check

0 steps flagged

No significant circularity; framework builds on standard graph-lift theory with independent simulation validation

full rationale

The paper defines dyadic matrices via signature row and recursive block structure, then maps lifted Tanner-graph cycles to tailless backtrackless closed walks on the protograph. This mapping is a direct consequence of established graph-lift theory once the protograph is fixed; the dyadic commutativity merely supplies an efficient enumeration algorithm without redefining the input. PEG-style construction uses forbidden shift sets to control girth or multiplicity, and absorbing-set enumeration follows from the same algebraic layout. Belief-propagation simulations are presented as external empirical checks on decoding gains, not as fitted parameters renamed as predictions. No self-citation chain is load-bearing for the central claims, no ansatz is smuggled via prior work, and no quantity is defined in terms of itself. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided summary.

pith-pipeline@v0.9.0 · 5556 in / 1036 out tokens · 32522 ms · 2026-05-08T19:29:34.192372+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 5 canonical work pages

  1. [1]

    Analysis of absorbing sets and fully absorbing sets of array-based LDPC codes,

    L. Dolecek, Z. Zhang, V. Anantharam, M. J. Wainwright, and B. Nikolic, “Analysis of absorbing sets and fully absorbing sets of array-based LDPC codes,”IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 181–201, 2009

    2009

  2. [2]

    Absorbing sets in quantum LDPC codes,

    K. D. Morris, T. Pllaha, and C. A. Kelley, “Absorbing sets in quantum LDPC codes,” Advances in Mathematics of Communications, vol. 23, no. 0, pp. 234–254, 2026. [Online]. Available: https://www.aimsciences.org/article/id/69a6ac251c9579521d08fdc7

    2026

  3. [3]

    Simple matrix-theory proof of the discrete dyadic convolution theorem,

    M. Gulamhusein, “Simple matrix-theory proof of the discrete dyadic convolution theorem,”Electronics Letters, vol. 9, pp. 238–239(1), May 1973. [Online]. Available: https://digital-library.theiet.org/content/journals/10.1049/el_19730172

    work page doi:10.1049/el_19730172 1973

  4. [4]

    Codes based on dyadic matrices and their generalizations,

    M. Martinez, T. Pllaha, and C. A. Kelley, “Codes based on dyadic matrices and their generalizations,”Advances in Mathematics of Communications, 2024. [Online]. Available: https://www.aimsciences.org/article/id/675c0e92d0b64a5d1d3497e9

    2024

  5. [5]

    A low complexity PEG-like algorithm to construct quasi-cyclic LDPC codes,

    A. Gómez-Fonseca, R. Smarandache, and D. G. M. Mitchell, “A low complexity PEG-like algorithm to construct quasi-cyclic LDPC codes,” in2023 12th International Symposium on Topics in Coding (ISTC), 2023, pp. 1–5

    2023

  6. [6]

    An efficient strategy to count cycles in the Tanner graph of quasi-cyclic LDPC codes,

    ——, “An efficient strategy to count cycles in the Tanner graph of quasi-cyclic LDPC codes,” IEEE Journal on Selected Areas in Information Theory, vol. 4, pp. 499–513, 2023

    2023

  7. [7]

    Regular and irregular progressive edge-growth Tannergraphs,

    X. Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressive edge-growth Tannergraphs,”IEEE Transactions on Information Theory, vol.51, no.1, pp.386–398, January 2005

    2005

  8. [8]

    Extremal absorbing sets in low-density parity-check codes,

    E. McMillon, A. Beemer, and C. A. Kelley, “Extremal absorbing sets in low-density parity-check codes,”Advances in Mathematics of Communications, vol. 17, no. 2, pp. 465–483, 2023

    2023

  9. [9]

    Quantum CSS LDPC codes with quasi-dyadic structure,

    A. Baldelli, M. Battaglioni, and P. Santini, “Quantum CSS LDPC codes with quasi-dyadic structure,” in2025 13th International Symposium on Topics in Coding (ISTC), 2025, pp. 1–5

    2025

  10. [10]

    Quantum error correction with girth-16 non-binary LDPC codes via affine permuta- tion construction,

    K. Kasai, “Quantum error correction with girth-16 non-binary LDPC codes via affine permuta- tion construction,” in2025 13th International Symposium on Topics in Coding (ISTC), 2025, pp. 1–5

    2025

  11. [11]

    Reproducible families of codes and cryptographic applications,

    P. Santini, E. Persichetti, and M. Baldi, “Reproducible families of codes and cryptographic applications,”Journal of Mathematical Cryptology, vol. 16, no. 1, pp. 20–48, 2022. [Online]. Available: https://doi.org/10.1515/jmc-2020-0003

    work page doi:10.1515/jmc-2020-0003 2022

  12. [12]

    Low-densityparity-check(LDPC)codesconstructedfromprotographs,

    J.Thorpe, “Low-densityparity-check(LDPC)codesconstructedfromprotographs,”Jet Propul- sion Laboratory Pasadena, CA, INP Progress Report 42-154, pp. 42–154, August 2003

    2003

  13. [13]

    A recursive approach to low complexity codes,

    R. M. Tanner, “A recursive approach to low complexity codes,”IEEE Transactions on Infor- mation Theory, vol. 27, no. 5, pp. 533–547, September 1981

    1981

  14. [14]

    Good quantum error-correcting codes ex- ist

    A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Physical Review A, vol. 54, pp. 1098–1105, Aug 1996. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevA.54.1098 30

    work page doi:10.1103/physreva.54.1098 1996

  15. [15]

    Simple quantum error-correcting codes,

    A. M. Steane, “Simple quantum error-correcting codes,”Physical Review A, vol. 54, pp. 4741–4751, Dec 1996. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.54.4741

    work page doi:10.1103/physreva.54.4741 1996

  16. [16]

    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, 2021

    2021

  17. [17]

    Quantum Kronecker sum-product low-density parity-check codes with finite rate,

    A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate,”Physical Review A, vol. 88, no. 1, p. 012311, 2013

    2013

  18. [18]

    Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength,

    J.-P. Tillich and G. Zémor, “Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength,”IEEE Transactions on Information Theory, vol. 60, no. 2, pp. 1193–1202, 2013

    2013

  19. [19]

    Quantum LDPC codes with almost linear minimum distance,

    P. Panteleev and G. Kalachev, “Quantum LDPC codes with almost linear minimum distance,” IEEE Transactions on Information Theory, vol. 68, no. 1, p. 213–229, Jan. 2022. [Online]. Available: http://dx.doi.org/10.1109/TIT.2021.3119384

    work page doi:10.1109/tit.2021.3119384 2022

  20. [20]

    Bounds on the minimum distance of linear codes and quantum codes,

    M. Grassl, “Bounds on the minimum distance of linear codes and quantum codes,” Online available at http://www.codetables.de, 2007, accessed on 2024-10-19

    2007

  21. [21]

    A necessary and sufficient condition for determining the girth of quasi-cyclic LDPC codes,

    X. Wu, X. You, and C. Zhao, “A necessary and sufficient condition for determining the girth of quasi-cyclic LDPC codes,”IEEE Transactions on Communications, vol. 56, no. 6, pp. 854–857, June 2008

    2008

  22. [22]

    Loop removal from LDPC codes,

    J. McGowan and R. Williamson, “Loop removal from LDPC codes,”IEEE Information Theory Workshop Proceedings, pp. 230–233, 2003

    2003

  23. [23]

    J. L. Gross and T. W. Tucker,Topological Graph Theory. USA: Wiley-Interscience, 1987

    1987

  24. [24]

    Design of irregular quasi-cyclic protograph codes with low error floors,

    R. Asvadi, A. H. Banihashemi, and M. Ahmadian-Attari, “Design of irregular quasi-cyclic protograph codes with low error floors,”IEEE International Symposium on Information Theory Proceedings, pp. 908–912, 2011

    2011

  25. [25]

    Lowering the error floor of LDPC codes using cyclic liftings,

    ——, “Lowering the error floor of LDPC codes using cyclic liftings,”IEEE Transactions on Information Theory, vol. 57, no. 4, pp. 2213–2224, April 2011

    2011

  26. [26]

    On codes designed via algebraic lifts of graphs,

    C. A. Kelley, “On codes designed via algebraic lifts of graphs,”2008 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 1254–1261, 2008

    2008

  27. [27]

    Message-passing algorithms for counting short cycles in a graph,

    M. Karimi and A. H. Banihashemi, “Message-passing algorithms for counting short cycles in a graph,”IEEE Transactions on Communications, vol. 61, no. 2, pp. 485–495, February 2013

    2013

  28. [28]

    A heuristic search for good low-density parity-check codes at short block lengths,

    Y. Mao and A. H. Banihashemi, “A heuristic search for good low-density parity-check codes at short block lengths,”IEEE International Conference on Communications Proceedings, vol. 1, pp. 41–44, June 2001

    2001

  29. [29]

    An algorithm for counting short cycles in bipartite graphs,

    T. R. Halford and K. M. Chugg, “An algorithm for counting short cycles in bipartite graphs,” IEEE Transactions on Information Theory, vol. 52, no. 1, pp. 287–292, January 2006

    2006

  30. [30]

    On absorbing sets of structured sparse graph codes,

    L. Dolecek, “On absorbing sets of structured sparse graph codes,” inInformation Theory and Applications Workshop (ITA), 2010, Jan 2010, pp. 1–5. 31

    2010

  31. [31]

    Towards improved LDPC code designs using absorbing set spectrum properties,

    L. Dolecek, J. Wang, and Z. Zhang, “Towards improved LDPC code designs using absorbing set spectrum properties,” in2010 6th International Symposium on Turbo Codes & Iterative Information Processing. IEEE, 2010, pp. 477–481

    2010

  32. [32]

    Absorbing sets of codes from finite geometries,

    A. Beemer, K. Haymaker, and C. A. Kelley, “Absorbing sets of codes from finite geometries,” Cryptography and Communications, vol. 11, no. 5, pp. 1115–1131, 2019

    2019

  33. [33]

    Analysis of syndrome-based iterative decoder failure of QLDPC codes,

    K. D. Morris, T. Pllaha, and C. A. Kelley, “Analysis of syndrome-based iterative decoder failure of QLDPC codes,” in2023 12th International Symposium on Topics in Coding (ISTC). IEEE, 2023, pp. 1–5

    2023

  34. [34]

    Reduced complexity iterative decoding of low- density parity check codes based on belief propagation,

    M. Fossorier, M. Mihaljevic, and H. Imai, “Reduced complexity iterative decoding of low- density parity check codes based on belief propagation,”IEEE Transactions on Communica- tions, vol. 47, no. 5, pp. 673–680, 1999

    1999

  35. [35]

    LDPC: Python tools for low density parity check codes,

    J. Roffe, “LDPC: Python tools for low density parity check codes,” https://pypi.org/project/ ldpc/, 2022, software package. 32

    2022