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A square array of pair partitions forces CSS orthogonality by linear equations, and a complete search algorithm certifies exact distances up to 24 in twenty quantum LDPC codes.

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2026-08-02 02:49 UTC pith:4WTF3WAT

load-bearing objection Genuinely new finite-length construction with clean proofs and a complete distance-verification search; the exact-distance table is credible but tied to an unrerun artifact, so accept with an independent rerun as a condition.

arxiv 2607.14091 v2 pith:4WTF3WAT submitted 2026-07-15 quant-ph

Pair-Partition Constructions for CPM-Based Quantum LDPC Codes

Koki Okada , Kenta Kasai This is my paper
classification quant-ph MSC 81P7094B0594B65
keywords quantum LDPC codesCalderbank–Shor–Steane (CSS) codescirculant permutation matricespair-partition constructionCSS orthogonalityminimum distance verificationgirth boundsquasi-cyclic codes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish a new method for building CSS quantum LDPC codes from circulant permutation matrices: choose a square array of pair partitions of the block-column types, impose linear paired-difference equations so that mixed X–Z overlaps cancel pairwise, and then screen the design with pairing graphs before searching exponents. It also proves a complete algorithm that, on a fixed CSS lift, returns NONE if and only if the quantum distance exceeds a given weight, making distance lower bounds machine-checkable certificates rather than heuristic evidence. The result would be a reproducible finite-length code family with certified parameters; the paper reports twenty exact-distance codes with distances 12–24 and rates up to 0.573, including five girth-eight codes. A supplementary permanent construction gives a factorial upper bound on distance under checkable rank and size conditions, so the distance claims are bracketed by proofs from both sides.

Core claim

The central claim is that CSS orthogonality for CPM-based codes reduces to a homogeneous linear condition: if the pair-partition array M pairs every block-column type and the exponents satisfy d_{j,u} - e_{i,u} = d_{j,v} - e_{i,v} for each pair {u,v} in M_{ij}, then H_X H_Z^T = 0 because the paired CPM contributions cancel over F2. This separates the algebraic validity of the code from its girth: pairing graphs derived from M give exponent-independent lower bounds on Tanner cycles (girth eight requires L ≥ 2J, and a universal girth-12 ceiling holds for the complete protograph). The paper further claims that Algorithm 2, which branches from unsatisfied checks over the CSS quotient and uses on

What carries the argument

The pair-partition array M is a J×J array of perfect matchings on Z_L; its paired-difference equations (2) are the identity that carries the argument, since they make mixed CPM differences cancel pairwise and turn CSS orthogonality into AM x = 0. Pairing graphs Γ^X_j and Γ^Z_i, built by colouring edges by the opposing index, turn cycle constraints into ordinary graph problems (parallel edges and triangles force short Tanner cycles), and Algorithm 2 is the complete distance verifier: it grows connected zero-syndrome supports outside rowspan G, uses as pruning lower bounds the number of unsatisfied checks divided by the maximum column weight and disjoint available-set counts, and is proved com

Load-bearing premise

The Table I numbers assume both the supplied program is bug-free and the binary ranks of the two parity-check matrices come out exactly to the maximum J(P−1)+1; if the program slips or any listed code fails that rank test, the reported distances and dimensions would not be established.

What would settle it

Re-run the posted software on the exact files matched by SHA-256 hashes: for each claimed exact-distance code, recompute rank H_X and rank H_Z and run Algorithm 2 through weight d−2 on both CSS sides. If any run returns a zero-syndrome vector outside the opposite row space at weight ≤ d−2, or if either rank differs from J(P−1)+1 while the theorem's conditions are claimed, the paper's Table I claims fail. Also re-check the explicit upper-bound vectors for zero syndrome and non-membership.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the paper is right, the paired-difference system reduces CSS orthogonality to a linear solve, so any CPM-based CSS construction satisfying (2) is automatically orthogonal; no separate search for commuting pairs is needed.
  • The pairing-graph design tests give a cheap pre-filter: arrays with repeated pairs or triangles can be discarded before any exponent search, and girth-eight targets require L ≥ 2J.
  • A completed run of Algorithm 2 is a proof that d > W, not a heuristic certificate; with explicit upper-bound vectors, equality d = w is established, making distance claims machine-checkable and reproducible for any fixed CSS lift.
  • For fixed column weight J, row weight L, and search limit W, the support-enumeration cost is bounded independently of the lift size P, so the verification algorithm scales more gracefully than a naive brute-force over all supports.
  • Under the checkable rank and size conditions, every PP code with column weight J has quantum distance at most (J+1)!, so the construction cannot produce distances above that for fixed J; growing distance requires growing J.

Where Pith is reading between the lines

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

  • The construction decouples commutation from girth in a way that might generalize: any pair partition satisfying the linear system yields an orthogonal CSS pair, so one could search the design space independently of the finite-field exponents, potentially opening larger parameter regions than the reported J≤4, L≤16 examples.
  • Algorithm 2 is not limited to CPM or PP codes; its correctness holds for any pair of binary matrices with rowspan G ⊆ ker H and a coordinate symmetry covering the roots, so it could certify distances of other structured quantum LDPC families provided a suitable automorphism group is known.
  • The factorial upper bound suggests that, for fixed J, distances cannot grow with lift size; to obtain asymptotically growing distance one needs J to grow, a direction the paper explicitly leaves untouched.
  • The cyclic-automorphism remark implies that reported logical representatives may not be independent logical classes; when designing logical gates or decoders, one should compute the induced action on the logical quotient, which the paper does not.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper introduces a pair-partition (PP) construction for CSS quantum LDPC codes built from circulant permutation matrices. The construction imposes linear paired-difference equations on the CPM exponent arrays, forcing H_X H_Z^T = 0 (Lemma 1/Theorem 1). Pairing graphs derived from the partition array are used to screen out arrays that force short Tanner cycles, and a permanent-based argument gives a universal upper bound d ≤ (J+1)! under rank and size conditions (Theorem 3). The central technical contribution is Algorithm 2, a complete search algorithm for verifying lower bounds on the CSS minimum distance; Theorem 2 proves that it returns NONE iff the distance exceeds the weight limit. The paper reports 20 exact-distance CPM-CSS codes and 2 codes with certified distance intervals (Table I), with distances up to 24 and girth up to 8.

Significance. If the numerical claims hold, the paper provides a useful systematic construction for finite-length CPM-based quantum LDPC codes, with exact distances reaching 24 and girth up to 8. The proof of Algorithm 2 is a solid and reusable contribution: it defines a correct pruning-based exhaustive search over the logical quotient, with a lift-size-independent combinatorial support-enumeration bound. The proofs of Lemma 1, Proposition 2, Lemma 2, and Theorem 2 are coherent and self-contained. The permanent upper bound (Theorem 3) is a nice complement. The paper also ships software and hashed run records, which is a strength; the numerical claims are conditional on the correctness of that artifact, but that is normal for computational quantum coding papers.

minor comments (4)
  1. [III, Proposition 3] The proof of the universal girth-12 upper bound is only a citation to Fossorier's cycle condition. Since the cancellation is not immediately obvious for two block rows and three block columns, please expand the argument to make the exponent-sum cancellation explicit.
  2. [V, Proposition 5] The L-root reduction is enabled under the stated conditions (prime P and W < P) in the implementation discussion, but Proposition 5 as stated does not list these hypotheses. Please state them explicitly in the proposition or its proof to avoid ambiguity about when the bound applies.
  3. [VI, parity discussion] Eq. (11) gives even parity for all fully populated CPM lifts, regardless of column weight. The phrase 'odd weights are impossible because every column has odd weight three' in the discussion of the girth-eight [ [1784,450,24] ] code is a special case; the same parity argument also underpins the J=4 claims such as [ [944,478,20] ]. A sentence clarifying this would prevent confusion.
  4. [VI and Reproducibility Materials] The exact-distance and interval claims rest on the correctness of the implementation of Algorithm 2 and the rank/girth checks. The paper provides code and hashed run records, which is good, but it would be helpful to describe how the implementation was validated (e.g., unit tests for row-space membership, cross-checks against small examples, or assertions in the search) and to specify exactly what the JSON run records certify. This would make the conditional nature of the numerical results easier for readers to verify.

Circularity Check

0 steps flagged

No significant circularity: the mathematical derivation is self-contained, and self-citations are contextual only.

full rationale

The paper's derivation chain is self-contained at the level of its mathematical claims. Lemma 1 proves H_X H_Z^T = 0 directly from the paired-difference equations (2): if {u,v} is in M_ij, then e_{iu} - d_{ju} = e_{iv} - d_{jv}, so the two CPM contributions cancel over F2. This is a direct implication of the stated constraint, not an equation used as its own conclusion. Theorem 2's completeness proof is explicit: it assumes only that a symmetry maps every support of weight at most W to a representative with least variable in the root set, proves Lemma 2 (connected minimum logical support), and shows the pruning bounds b1 and b2 are valid. It explicitly disclaims reliance on the cited stopping-set literature: 'The preceding proof is self-contained: the cited classical low-weight search literature motivates the branching strategy but is not used to justify the distance lower bound.' Proposition 4 establishes the cyclic symmetry used to reduce the root set, so the L-root reduction is proved rather than assumed. Theorem 3's factorial upper bound uses the standard maximal-minor identity over F2 and cites the external bound [25] for the weight estimate; its rank assumption is stated and verified per code, not derived from the conclusion. The self-citations [9]-[16] appear only as related-work context, and the paper explicitly says the bank of forbidden patterns 'cannot certify distance' and that 'Distance is certified only by a complete search on the fixed matrices.' Thus no prediction is obtained by fitting a parameter; exact distances are certified by exhaustive search plus explicit witnesses. The only non-mathematical dependency is the correctness of the provided software implementation of Algorithm 2, which is a reproducibility/verification issue rather than a circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No numeric constants are fitted; P, J, L, the pair-partition array, and the exponents are search outputs, not fitted parameters. The central claims rest on standard linear algebra, the CSS construction, Fossorier's cycle condition, and the external permanent-weight bound [25].

axioms (8)
  • standard math CSS orthogonality H_X H_Z^T = 0 yields a valid stabilizer code with dimension n - rank H_X - rank H_Z (standard CSS construction).
    Invoked in Theorem 1 to convert the orthogonality condition into a code dimension; background from [3],[4].
  • standard math Fossorier's CPM cycle condition: a length-2r Tanner cycle exists iff a signed alternating sum of CPM exponents is zero.
    Used in Propositions 2-3 and in the 4/6-cycle screening of Section III; external theorem from [7].
  • standard math Mantel's theorem: a triangle-free graph on L vertices has at most L^2/4 edges.
    Used in Corollary 1 to derive L >= 2J for girth eight.
  • standard math For odd prime P, R_P = F2[x]/(x^P-1) decomposes via Chinese remainder into F2[x]/(x-1) times fields for each nontrivial irreducible factor.
    Used in Theorem 3 to count kernel dimensions of the polynomial parity-check matrices.
  • standard math Maximal-minor (permanent/determinant in char 2) vectors of a rank-J matrix span its kernel; Laplace/Plücker identities give bHZ c_S^T = 0.
    Core of the factorial upper bound proof in Theorem 3.
  • standard math Smarandache-Vontobel bound: a permanent of a J x J matrix has at most J! monomials, so c_S has weight at most (J+1)!.
    Used to bound the weight of logical witnesses in Theorem 3; external result [25].
  • domain assumption For a fully populated J x L CPM matrix, the J block-row families each sum to the all-one vector, so binary rank <= J(P-1)+1.
    Proved in the text and assumed for the stated rank equality in Theorem 3; verified for all Table I codes.
  • domain assumption The lift size P is an odd prime and the search requires W < P for the cyclic-root reduction to be complete.
    Assumed in the setup (Section II) and in the implementation note before Proposition 5.

pith-pipeline@v1.3.0-alltime-deepseek · 14979 in / 34070 out tokens · 310129 ms · 2026-08-02T02:49:31.991999+00:00 · methodology

0 comments
read the original abstract

We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.

Figures

Figures reproduced from arXiv: 2607.14091 by Kenta Kasai, Koki Okada.

Figure 1
Figure 1. Figure 1: A 3 × 3 pair-partition array on Z8 used in Example 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The full 36 × 48 coefficient matrix AM obtained from [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pairing graphs obtained from the array in Example [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum LDPC codes with design rate 1/5 and good performance below 1000 physical qubits

    quant-ph 2026-07 conditional novelty 7.0

    A new family of rate-1/5 quantum LDPC codes built from non-abelian group symmetries approaches teraquop-region memory error rates below 1000 physical qubits under an optimistic, extrapolated benchmark.

Reference graph

Works this paper leans on

27 extracted references · 6 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Low-density parity-check codes,

    R. G. Gallager, “Low-density parity-check codes,”IRE Transactions on Information Theory, vol. 8, no. 1, pp. 21–28, Jan. 1962

    1962

  2. [2]

    A recursive approach to low complexity codes,

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

    1981

  3. [3]

    Good quantum error-correcting codes exist,

    A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Physical Review A, vol. 54, pp. 1098–1105, 1996

    1996

  4. [4]

    Error correcting codes in quantum theory,

    A. M. Steane, “Error correcting codes in quantum theory,”Physical Review Letters, vol. 77, pp. 793–797, 1996

    1996

  5. [5]

    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

    2004

  6. [6]

    Quantum low-density parity-check codes,

    N. P. Breuckmann and J. N. Eberhardt, “Quantum low-density parity-check codes,”PRX Quantum, vol. 2, p. 040101, 2021

    2021

  7. [7]

    Quasicyclic low-density parity-check codes from circulant permutation matrices,

    M. P. C. Fossorier, “Quasicyclic low-density parity-check codes from circulant permutation matrices,”IEEE Transactions on Information Theory, vol. 50, no. 8, pp. 1788–1793, Aug. 2004

    2004

  8. [8]

    Quantum quasi-cyclic LDPC codes,

    M. Hagiwara and H. Imai, “Quantum quasi-cyclic LDPC codes,” inProceedings of the 2007 IEEE International Symposium on Information Theory, 2007, pp. 806–810

    2007

  9. [9]

    Explicit construction of classical and quantum quasi-cyclic low-density parity-check codes with column weight 2 and girth 12,

    D. Komoto and K. Kasai, “Explicit construction of classical and quantum quasi-cyclic low-density parity-check codes with column weight 2 and girth 12,” arXiv:2501.13444, 2025

    Pith/arXiv arXiv 2025

  10. [10]

    Quantum error correction near the coding theoretical bound,

    ——, “Quantum error correction near the coding theoretical bound,”npj Quantum Information, vol. 11, p. 154, 2025. [Online]. Available: https://www.nature.com/articles/s41534-025-01090-1

    2025

  11. [11]

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

    K. Kasai, “Quantum error correction with girth-16 non-binary LDPC codes via affine permutation construction,” arXiv:2504.17790, 2025

    Pith/arXiv arXiv 2025

  12. [12]

    Breaking the orthogonality barrier in quantum LDPC codes,

    ——, “Breaking the orthogonality barrier in quantum LDPC codes,” arXiv:2601.08824, 2026

    arXiv 2026

  13. [13]

    High-girth regular quantum LDPC codes from affine-coset structures,

    K. Okada and K. Kasai, “High-girth regular quantum LDPC codes from affine-coset structures,” arXiv:2604.20838, 2026

    Pith/arXiv arXiv 2026

  14. [14]

    High-girth regular quantum LDPC codes from square-base hypergraph products via CPM lifts,

    ——, “High-girth regular quantum LDPC codes from square-base hypergraph products via CPM lifts,” arXiv:2604.27817, 2026

    Pith/arXiv arXiv 2026

  15. [15]

    A two-branch finite-field construction for regular CSS LDPC bases,

    ——, “A two-branch finite-field construction for regular CSS LDPC bases,” arXiv:2605.23894, 2026

    Pith/arXiv arXiv 2026

  16. [16]

    Rate-2/3 girth-8 (3,18)-regular quantum LDPC codes from two-branch finite-field bases and CPM lifts,

    ——, “Rate-2/3 girth-8 (3,18)-regular quantum LDPC codes from two-branch finite-field bases and CPM lifts,” arXiv:2606.27130, 2026

    Pith/arXiv arXiv 2026

  17. [17]

    On the iterative decoding of sparse quantum codes,

    D. Poulin and Y . Chung, “On the iterative decoding of sparse quantum codes,”Quantum Information & Computation, vol. 8, no. 10, pp. 987–1000, Nov. 2008

    2008

  18. [18]

    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

  19. [19]

    Decoding across the quantum low-density parity-check code landscape,

    J. Roffe, D. R. White, S. Burton, and E. T. Campbell, “Decoding across the quantum low-density parity-check code landscape,”Physical Review Research, vol. 2, p. 043423, 2020

    2020

  20. [20]

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

    J.-P. Tillich and G. Zemor, “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, Feb. 2014

    2014

  21. [21]

    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, pp. 213–229, Jan. 2022

    2022

  22. [22]

    Finding small stopping sets in the tanner graphs of LDPC codes,

    G. Richter, “Finding small stopping sets in the tanner graphs of LDPC codes,” inProceedings of the 4th International Symposium on Turbo Codes and Related Topics and the 6th International ITG-Conference on Source and Channel Coding, 2006, pp. 1–5

    2006

  23. [23]

    An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices,

    E. Rosnes and Ø. Ytrehus, “An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices,”IEEE Transactions on Information Theory, vol. 55, no. 9, pp. 4167–4178, Sep. 2009

    2009

  24. [24]

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

    S. M. Khatami, L. Danjean, D. V . Nguyen, and B. Vasi ´c, “An efficient exhaustive low-weight codeword search for structured LDPC codes,” inProceedings of the 2013 Information Theory and Applications Workshop, 2013, pp. 401–410

    2013

  25. [25]

    Quasi-cyclic LDPC codes: Influence of proto- and tanner-graph structure on minimum hamming distance upper bounds,

    R. Smarandache and P. O. V ontobel, “Quasi-cyclic LDPC codes: Influence of proto- and tanner-graph structure on minimum hamming distance upper bounds,”IEEE Transactions on Information Theory, vol. 58, no. 2, pp. 585–607, Feb. 2012. 16

    2012

  26. [26]

    Pair-partition constructions for CPM-based quantum LDPC codes,

    K. Okada and K. Kasai, “Pair-partition constructions for CPM-based quantum LDPC codes,” https://kasai.ict.eng.isct.ac.jp/pair_partition_ cpm_css_codes_20260714.html, 2026, construction data page, accessed July 28, 2026

    2026

  27. [27]

    Pair-partition CPM CSS code construction software,

    ——, “Pair-partition CPM CSS code construction software,” https://github.com/kasaikenta/pair-partition-cpm-css-codes, 2026, gitHub repository, accessed July 30, 2026

    2026