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Pair-partition equations guarantee CSS orthogonality and yield 34 quantum LDPC codes with certified exact distances.

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2026-08-04 01:38 UTC pith:4WTF3WAT

load-bearing objection Solid construction plus a provably complete distance-verification routine; exact-distance table rests on external certificates rather than anything in the manuscript, but the math holds. the 2 major comments →

arxiv 2607.14091 v3 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 PACS 03.67.Pp
keywords quantum LDPC codesCSS codescirculant permutation matricespair-partition constructiongirthminimum distance verificationquasi-cyclic codesexact distance
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 introduces a construction method for binary CSS quantum LDPC codes in which a square array of pair partitions enforces linear paired-difference equations on the exponents of circulant permutation matrices, making X–Z orthogonality automatic. It proves a complete search algorithm that certifies a quantum-distance lower bound by exhausting zero-syndrome vectors outside the opposing stabilizer row space, and reports thirty-four codes whose minimum distances are established exactly, including seven with Tanner girth eight and distances up to 24. The construction stage is screened by pairing graphs before any exponent search, and the verification stage is a directly checkable finite procedure. If the claims hold, the paper supplies a reproducible route to finite-length quantum LDPC codes with known distances, which matters for practical decoders and for benchmarking against code families without exact distance guarantees.

Core claim

The authors claim that any pair-partition array M, together with exponent arrays satisfying the paired-difference equations, produces a valid CPM-based CSS code, and they prove that the paired terms cancel over F2 to give H_X H_Z^T = 0. They further prove that their Algorithm 2 is complete: for matrices with the stated cyclic symmetry, it returns NONE if and only if the corresponding distance bound d(H,G) exceeds the search weight W. Under the natural full-rank and size conditions, they show via maximal-minor permanent vectors that any such code has a logical operator of weight at most (J+1)!, giving a lift-size-independent upper bound. The reported list of thirty-four exact-distance regular

What carries the argument

The central object is the pair-partition array M=(M_ij), where each entry is a partition of the L block-column types into pairs; the paired-difference equations d_{j,u}-e_{i,u}=d_{j,v}-e_{i,v} for each pair (u,v) force the mixed X–Z differences to match, so identical CPMs cancel in pairs and CSS orthogonality becomes a linear condition. The pairing graphs Γ^X_j and Γ^Z_i, formed by the matchings in M, let the design reject forced short cycles before exponent search. The proof of distance uses Algorithm 2, a complete DFS search on the quotient kerH \ rowspanG with provable pruning bounds and L cyclic roots replacing LP roots, together with Fossorier's CPM cycle condition and permanent-based m

Load-bearing premise

The exact-distance entries in Table I rely on the posted SHA-256-identified run records and the released software faithfully implementing Algorithm 2; if those artifacts are incomplete or the implementation diverges from the pseudocode at some branch, the table's distance column is not established.

What would settle it

Pick any listed code, say [[944,478,20]], and rerun the posted verification implementation on the posted fixed matrices through weight 18 on both CSS sides. If any zero-syndrome vector outside the opposing stabilizer row space appears at weight 18 or below, the lower-bound claim is false; independently, checking a claimed weight-20 logical vector for zero syndrome and non-membership in the opposite row space would falsify the upper bound if it fails.

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

If this is right

  • Exact minimum distances for fixed finite CPM-based CSS codes become verifiable by an exhaustive, symmetry-reduced search rather than by heuristics or extrapolation.
  • The pairing-graph screening gives a fast, lift-size-independent way to discard designs that cannot reach girth eight, since girth eight requires L ≥ 2J and forces triangle-free simple pairing graphs.
  • Under the stated rank and size conditions, every such code is guaranteed to contain a logical operator of weight at most (J+1)!, complementing the instance-specific lower bounds.
  • Each reported code comes with posted verification records, so independent reruns of the proved algorithm can confirm or refute the table's distance and rank claims.
  • The construction separates commutation (from the pairing array) from girth and distance (from the completed lift), so each stage can be optimized and checked in isolation.

Where Pith is reading between the lines

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

  • The L-root cyclic reduction in Algorithm 2 likely extends to any CSS lift admitting a transitive coordinate symmetry that preserves both kernel and row space, potentially enabling exact distance searches in other quasi-cyclic families.
  • Because the reported codes have fixed (J,L), the construction does not by itself yield growing distance; a natural next step is to search for sequences of pair-partition arrays whose pairing graphs survive the girth tests while rank conditions remain checkable.
  • The factorial upper bound counts monomials before cancellation; a refined analysis of permanent cancellations might lower the (J+1)! bound for specific exponent choices.
  • The pairing-graph design screen could be reused as a preprocessing filter in any CPM-based code construction, even when the paired-difference equations are not the final constraint.

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

2 major / 5 minor

Summary. The paper introduces a pair-partition (PP) construction for binary CSS quantum LDPC codes built from circulant permutation matrices. A J×J array of pair partitions of the block-column index set imposes linear paired-difference constraints on the CPM exponent arrays, forcing H_X H_Z^T = 0 (Lemma 1, Theorem 1). The construction is separated into a combinatorial pre-screening stage using pairing graphs (Propositions 1–3, Corollary 1) and an exponent/full-lift search stage. The main verification contribution is Algorithm 2, a complete branch-and-bound search for minimum-weight vectors in the CSS quotient ker H \ rowspan G; Theorem 2 states that it returns NONE iff d(H,G) > W. The paper also proves a universal factorial upper bound d ≤ (J+1)! under full-rank and size conditions (Theorem 3). The numerical result is a table of thirty-four exact-distance regular CSS codes with parameters, girth, lift sizes, and rates, with distances certified as matching lower and upper bounds.

Significance. If the numerical claims hold, the paper provides a useful finite-length construction method and a provably correct distance-verification procedure for CPM-based CSS codes. The mathematical core is coherent and self-contained: the CSS orthogonality proof, the pairing-graph girth obstructions, the completeness proof of Algorithm 2 (including the b1/b2 pruning bounds), and the factorial upper bound all appear sound. I also credit the authors for separating lower-bound certificates from upper-bound witnesses, for proving that the cyclic-root reduction preserves completeness, and for making the supporting files and code available with SHA-256 hashes. The significance is limited by the fact that the exact-distance table rests on external verification records that are not fully inspectable in the manuscript; the mathematical framework, however, is not affected by that evidential gap.

major comments (2)
  1. [Section VI, Table I; Reproducibility Materials] The central numerical claim—the thirty-four exact-distance codes—is established only through records that are not in the manuscript: SHA-256-hashed JSON shards, matrix parameters, and a GitHub implementation. Theorem 2 proves the logical correctness of Algorithm 2, but a completed 'NONE' is a certificate only if the implementation matches the pseudocode at every branch and if every shard is complete and correctly hashed. The manuscript's phrasing that entries were 'independently verified' appears to mean reruns of the same code and records, not an independent implementation. Because Table I is the paper's main deliverable, the certificates need to be part of the journal's archival supplementary material (not only a project webpage), with the exact repository commit and a specification of the verification environment. Please include at least one fully expanded certificate in the manuscrip
  2. [Section VI, rows for [[492,170,20]], [[516,178,20]], [[2110,848,22]], [[2230,896,24]], [[944,478,20]]] Several exact-distance entries rely on transferring a complete lower bound from one CSS side to the other using maps described only as 'a verified weight-preserving involution' or 'a verified CPM-affine coordinate isomorphism'. These maps are never specified in the text. Since the equality d_X = d_Z and hence d = min{d_X,d_Z} depends on them, the maps should be given explicitly (or at least included in the per-code verification metadata) so that a reader can check that they exchange the two row spaces and the two logical quotients. This is a reproducibility gap in the lower-bound proof chain, not a flaw in the underlying search algorithm.
minor comments (5)
  1. [Section V.A, Corollary 2] The phrase 'these P binary vectors have pairwise distinct supports' could be misread as 'pairwise disjoint supports.' The proof only establishes that no two are equal. Please reword to 'no two are equal as sets' to avoid potential confusion, especially since the later orbit arguments do not require disjointness.
  2. [Algorithm 2, Section V.A] The root-coverage assumption in Theorem 2 is stated abstractly. For CPM matrices it is justified in the following paragraph, but the proof would be easier to follow if the argument that translating a selected coordinate in the least block column to (ℓ0,0) makes that variable the least in the support were stated explicitly at the point of the theorem.
  3. [Section VI] The per-code explanatory paragraphs are somewhat repetitive. Consider replacing the prose with a table column or a short legend indicating, for each row, the lower-bound method (complete X-search, complete Z-search, symmetry transfer) and the upper-bound witness (explicit vector, permanent vector). This would make the verification structure of Table I more readable without losing information.
  4. [Section IV.C, Example 4] The example says the search 'forms the three parity tables q_{j,b}(D)' for J=3. If the same pattern bank is intended for general J, it would help to state explicitly how q is generalized to arbitrary J; currently the example is tied to J=3.
  5. [General] The distinction between the complete Algorithm 2 and the heuristic candidate-screening loop in Algorithm 1 is clear, but the phrase 'independent verification' in Section VI should be defined at first use: does it mean an independent implementation, a rerun on a different machine, or a re-check of the existing shards? A precise definition would prevent ambiguity.

Circularity Check

0 steps flagged

No circularity: the PP construction and Algorithm 2 distance verification are self-contained; self-citations are contextual only.

full rationale

The paper's derivation chain is self-contained rather than circular. CSS orthogonality is derived directly from the paired-difference equations (2) in Lemma 1 and Theorem 1, with no hidden dependence on the claimed code list. The pairing-graph girth conditions (Propositions 1 and 2, Corollary 1) follow from elementary graph counting and Fossorier's CPM cycle condition, an external standard result. The central distance-verification claim, Theorem 2, is proved from first principles: it establishes that Algorithm 2 returns NONE iff d(H,G) > W using the invariant S subset of supp(x), the parity argument at unsatisfied checks, and the valid pruning bounds b1 and b2. The proof does not invoke the authors' prior work, and the paper explicitly states that the cited low-weight search literature 'motivates the branching strategy but is not used to justify the distance lower bound.' The forbidden-pattern bank is explicitly disclaimed as a distance proof: 'stored patterns accelerate candidate generation but do not prove distance' and 'The bank B contains only previously observed supports. Passing it does not exclude all low-weight logical operators and is not a distance proof.' Upper bounds come from explicit non-stabilizer zero-syndrome vectors and from Theorem 3's permanent-construction argument, which rests on standard linear algebra and the rank/size conditions verified for the reported codes. Self-citations [9]-[16] appear only as contextual references to related CPM constructions; they are not load-bearing for the new construction, the completeness theorem, or the numerical claims. The SHA-256-hashed external verification records and GitHub implementation are reproducibility artifacts rather than inputs to the mathematical derivation; any concern about their integrity is an empirical/data-availability matter, not circularity. No step in the paper reduces, by construction or by definition, to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No free parameters are fitted to data; J, L, P, and W are explicit design/search parameters stated for every code and theorem, not concealed degrees of freedom. The axioms are standard mathematical facts or verified conditions rather than ad hoc constructs introduced to force the results.

axioms (7)
  • domain assumption CSS orthogonality H_X H_Z^T = 0 gives a valid stabilizer code with dimension k = LP - rank H_X - rank H_Z.
    Standard CSS construction invoked in Section II-C and used to compute k for every code in Table I.
  • domain assumption Fossorier's CPM cycle conditions characterize Tanner cycles in quasi-cyclic matrices.
    Used in Propositions 2-3 and in the exponent-level cycle tests of Section III; cited from [7].
  • standard math Over characteristic two, permanents equal determinants, and the maximal-minor vectors span the kernel of a full-row-rank matrix.
    Foundation of the permanent construction in Theorem 3, Eq. (12), and the dimension computation of C_Z.
  • domain assumption For the pairs (H,G) used in the distance search, rowspan G is contained in ker H.
    Holds for (HX,HZ) and (HZ,HX) because of CSS orthogonality; required for d(H,G) in Eq. (9) and for Theorem 2.
  • domain assumption The rank equality rank H_X = rank H_Z = J(P-1)+1 holds for every code in Table I.
    Verified computationally for each reported code; it is the natural maximum-rank condition used in Theorem 3. It is not proven from the PP construction alone.
  • standard math The cyclic shift T preserves ker H and rowspan G for CPM lifts.
    Proved in Proposition 4 and required for the L-root reduction in the complete distance search and for Theorem 3's shift-invariance argument.
  • standard math For odd prime P, x^P - 1 is square-free over F2 and the Chinese-remainder decomposition into fields applies.
    Used in the CRT argument of Theorem 3 to compute dimensions of rank and kernel spaces in nontrivial components.

pith-pipeline@v1.3.0-alltime-deepseek · 16479 in / 17565 out tokens · 157198 ms · 2026-08-04T01:38:27.539321+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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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

    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