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REVIEW 4 major objections 4 minor 36 references

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A divisor-driven search for bicycle quantum LDPC codes finds $[[66,20,7]]_2$ with $kd^2/n = 14.85$, above the bivariate bicycle $[[144,12,12]]_2$ at less than half the block length, and proves $[[48,10,5]]_2$ impossible in the weight-8…

desk verdict Useful finite-length search framework with a correct gcd dimension formula (despite a flawed proof step), but the headline distances and the n=48 exclusion rest on unshipped computations. read the letter →

arxiv 2608.09115 v1 pith:34ICNSKZ submitted 2026-08-10 cs.IT math.ITquant-ph

Liangdong Lu , Guanmin Guo , Yang Liu , Ruipan Yang This is my paper · ORCID
classification cs.ITmath.ITquant-ph MSC 94B0594B3581P70
keywords quantumLDPCcodesbicyclestabilizercyclicCalderbankcorrespondencepolynomialgcdexactdistancecoset-2BGA
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that bicycle quantum LDPC codes, in the cyclic case, can be reformulated entirely in the polynomial ring $\mathbb{F}_2[x]/(x^l-1)$, where self-orthogonality is automatic, the dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\mathbb{F}_4$ with the stabilizer excluded. This turns code search into an algebraically pre-filtered enumeration that reaches regimes poorly covered by existing tables, recovering the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and finding a family of high-$kd^2/n$ codes, including $[[66,20,7]]_2$ with $kd^2/n = 14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n = 12$) at less than half the block length. The same framework delineates its own boundary: an exhaustive census at $n=48$ exhibits a $[[48,10,6]]_2$ code from a minimal 48-element group, and proves that distance $5$ forces a stabilizer-rank loss, excluding $[[48,10,5]]_2$ from the weight-8 symmetric coset-2BGA family. If the claimed computations are correct, the paper provides a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, with explicit polynomials for each code.

What carries the argument

The load-bearing object is the cyclic polynomial model: circulant blocks become multiplication by $a(x)$, $b(x)$ in $R = \mathbb{F}_2[x]/(x^l-1)$, with the involution $a(x) \mapsto a(x^{-1})$ playing the role of transposition. Self-orthogonality is automatic because the ring is commutative, so the CRSS condition reduces to the identity $a(x)b(x)+b(x)a(x)=0$. The dimension formula $k = 2\deg\gcd(a, b, x^l-1)$ (Proposition 4) gives algebraic dimension control before any matrix is built, and Theorem 7 shows that candidates $a = gu$, $b = gv$ with $g$ a divisor of $x^l-1$ dial the dimension: $k = 2\deg g$ whenever $u$ and $v$ share no cyclotomic factor of $(x^l-1)/g$. The exact distance is computed by Lemma 10 as the smallest weight of a kernel vector that lies outside the stabilizer row space, with the Calderbank correspondence ensuring this matches the quantum minimum distance. The search pipeline applies the gcd filter, a fingerprint dedup, and an involution-symmetry test before building matrices, so the expensive weight-enumeration runs only on candidates that already pass algebraic pre-filters.

What would settle it

Recomputing the exact distance of the $[[66,20,7]]_2$ code from the supports in Table II with the stabilizer excluded and finding $d < 7$ would refute the central parameter claim; independently, finding a single $[[48,10,5]]_2$ code inside the weight-8 symmetric coset-2BGA family at $n=48$ would refute Theorem 16. The census itself can be checked by re-enumerating the 123 equivalence classes and verifying that every $d=5$ class has $\operatorname{rank} H_X=19$, $\operatorname{rank} H_Z=20$.

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Extended reading notes

Core claim

The discovery is that the cyclic bicycle construction collapses into the polynomial ring $\mathbb{F}_2[x]/(x^l-1)$: the two circulant blocks become multiplication by $a(x)$ and $b(x)$, self-orthogonality follows from commutativity of the ring, and the quantum dimension equals $2\deg\gcd(a,b,x^l-1)$, so $k$ is known before any matrix is built (Proposition 4). The distance is computed exactly as the minimum weight of a vector in $\ker H_Z \setminus \operatorname{row} H_X$ (and symmetrically), using the Calderbank correspondence, with the stabilizer excluded to avoid undercounting (Lemma 10). A divisor-driven search that writes $a = gu$, $b = gv$ with $g \mid x^l-1$ finds, among others, $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$, $[[66,20,7]]_2$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, and $[[90,20,6]]_2$, recovering $[[42,12,4]]_2$ and $[[62,12,4]]_2$; the $[[66,20,7]]_2$ code attains $kd^2/n = 14.85$. Stepping outside the cyclic case, the paper shows at $n=48$ that the weight-8 symmetric coset-2BGA family contains no $[[48,10,5]]_2$ code—distance $5$ forces ranks $19$ and $20$ and hence $k=9$—while a $[[48,10,6]]_2$ code is realized by a minimal 48-element group, one third the order of the earlier 72-element realization of the same parameters.

Load-bearing premise

The argument rests on the computer-assisted exact distances reported in Tables I-V and on the exhaustive 123-class census behind Theorem 16, and the preprint provides no code, data files, or hashes for an independent reader to audit those computations.

Editorial extensions

If this is right

  • The cyclic search reduces candidate filtering to a gcd computation, so exploring new lengths $l$ mainly requires factoring $x^l-1$ and enumerating divisors, a much smaller search than building all circulant pairs.
  • The $[[66,20,7]]_2$ code outperforms the bivariate bicycle $[[144,12,12]]_2$ in $kd^2/n$ at less than half the block length, so high-rate, moderate-distance bicycle codes are available at short lengths.
  • The $n=48$ census proves that within the weight-8 symmetric coset-2BGA family, distance $5$ and dimension $10$ are mutually exclusive, giving a concrete boundary between cyclic-polynomial and genuinely coset-theoretic phenomena.
  • The $[[48,10,6]]_2$ code realized by a 48-element group shows that the same public parameters can be obtained with a smaller group than the 72-element realization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the exact-distance computations hold, the same divisor-search pipeline should extend to structured divisors at larger lengths, potentially producing codes with $kd^2/n$ above lifted-product families; a natural next check is the two-level estimator the paper proposes, in which a decoder proposes low-weight logical candidates and exact certification confirms them.
  • The rank-degeneracy theorem suggests a wider pattern: in coset-2BGA families, high-distance strata may systematically force a rank defect and hence dimension loss; testing other indices $[G:H] = m$ could reveal where the cyclic gcd formula's defect-zero picture fails.
  • The $k=2$ family built from $g = x+1$ has distance growing roughly with block length while stabilizer weight stays at most $16$; if that trend persists, it yields a simple systematic family of low-rate, high-distance bicycle codes worth circuit-level threshold analysis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops a polynomial-ring reformulation of two-block circulant (bicycle) quantum LDPC codes over F2, working in R = F2[x]/(x^l-1). It claims that self-orthogonality is automatic in this cyclic setting, the quantum dimension is given by the gcd formula k = 2 deg gcd(a,b,x^l-1), and the quantum distance can be certified exactly by a Calderbank-correspondence computation that excludes the stabilizer row space. On this basis the authors run a divisor-driven search, reporting many new codes with competitive kd^2/n, notably [[66,20,7]]_2 with kd^2/n = 14.85, together with a family of k=2 codes and several n=90 codes. They also move outside the cyclic case to a weight-8 symmetric coset-2BGA family at n=48, presenting a [[48,10,6]]_2 code from a 48-element group and a computer-assisted theorem excluding [[48,10,5]]_2 from that family via a stabilizer-rank degeneracy.

Significance. If the computational claims are correct, this is a genuinely useful contribution: it gives an algebraic pre-filter for bicycle-code search, a simple parameter-free dimension formula, exact distance certification rather than heuristic bounds, and a first systematic statement of where coset-theoretic phenomena begin. The paper ships explicit polynomial supports for all headline codes in Tables II and III, and the decoding study is decoder-fair with clear simulation parameters. The main weakness is audibility: the exact-distance computations behind Tables I-V and the exhaustive 123-class census behind Theorem 16 are not reproducible from the preprint, and the proof of Proposition 4 contains a false injectivity assertion. These issues are fixable but are load-bearing for the paper's central existence and nonexistence claims.

major comments (4)
  1. [Section III-C, Proposition 4] ok
  2. [Section IV-B and Tables I-V] ok
  3. [Section VII, Theorem 16] ok
  4. [Section III-A, Lemma 1 and Remark 3] ok
minor comments (4)
  1. [Section V, paragraph after Table III] ok
  2. [Section V, Table III caption] ok
  3. [Section VI, Table VII] ok
  4. [Section VIII-C] ok

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: dimension is a proven algebraic identity and distances are exact computations, not fitted quantities.

full rationale

The paper's central chain is: (i) a polynomial reformulation of cyclic bicycle codes with automatic self-orthogonality (Theorem 2), (ii) a derived dimension formula k=2 deg gcd(a,b,x^l-1) (Proposition 4) together with a divisor presentation theorem (Theorem 7), and (iii) an exact-distance computation via the two-kernel CSS check (Lemma 10), with a search that uses the gcd dimension as a pre-filter and distances as verification. No fitted parameter is promoted to a prediction: k is set by the chosen divisor g and then cross-checked by rank; d is computed exactly by kernel enumeration rather than inferred from a model. The comparison figure kd^2/n is evaluated directly from the verified parameters. The paper contains no self-citations, so the self-citation and imported-uniqueness patterns do not apply. The n=48 boundary theorem is computer-assisted and depends on an exhaustive census whose full data are deferred to a companion paper; that is a reproducibility and auditability concern, not circularity. The invalid containment step in the proof of Proposition 4 is a proof gap, not a circular step, since the formula is independently derivable and is not used to define the codes. The derivation is therefore self-contained: inputs (polynomial pairs) are not defined in terms of outputs (codes and parameters), and no claimed result reduces by construction to its own assumptions.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central existence claims rest on standard cyclic-code algebra plus computational exact-distance and census procedures. The main unshipped inputs are the Magma implementation and the n=48 exhaustive census data. No new physical entities are introduced.

free parameters (2)
  • Decoding curve fit parameters (alpha, beta, gamma) = Per-code values in Table VII, e.g. [[42,6,7]]: (12.50, -88, 359)
    Fitted in Eq. (2) to Monte Carlo logical error rates; used for pseudo-threshold extrapolation and plots, not for the existence or distance claims.
  • Search campaign thresholds (d_min, enumeration cap T, l-list) = d_min=5, T=2e5, l in {21,23,25,27,31,33,35,39,45}
    Hand-set exploration budget. The headline codes are existence results, but any claim of exhaustiveness at n=48 or of best code found at each k is conditioned on these thresholds.
assumptions (5)
  • standard math CRSS correspondence: symplectic self-orthogonal binary codes are additive self-orthogonal codes over F4, and quantum distance is the minimum weight of the coset C_s^perp \ C.
    Invoked in Section II-B and used for all exact distance computations through Lemma 10.
  • standard math For odd l, R=F2[x]/(x^l-1) is squarefree and a principal ideal ring; the rank of the joint row space of (a,b) equals l - deg gcd(a,b,x^l-1).
    Underlies Proposition 4 and the gcd pre-filter. The paper's proof has a false injectivity step, but the statement follows from the CRT product decomposition.
  • ad hoc to paper The exact distance computation of Lemma 10 as implemented in Magma correctly enumerates kernel weight classes and checks membership in the stabilizer row space.
    All headline exact distances depend on this implementation; no code or data are released to verify it.
  • ad hoc to paper The exhaustive n=48 coset-2BGA census is complete, and the 123 equivalence classes and rank/weight spectra are computed correctly.
    Theorem 16 is computer-assisted and only sketched; the full data are promised in a companion paper, not shipped here.
  • domain assumption BP-OSD with 50 sum-product iterations and OSD-CS order 1 is an appropriate decoder under code-capacity depolarizing noise.
    Used for the decoding benchmarks of Section VI, following the setup of the BB and trivariate bicycle references.

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Pith. "Pith review of Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search." pith.science (2026). https://pith.science/paper/34ICNSKZ

@misc{pith2026260809115,
  author       = {Pith},
  title        = {Pith review of: Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34ICNSKZ}},
  note         = {Machine review of arXiv:2608.09115}
}
abstract

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

Figures

Figures reproduced from arXiv: 2608.09115 by the authors.

Figure 1
Figure 1. Parameter landscape at n = 66 and n = 90: exact distance d against dimension k for the best code found at each k; labels give kd2/n (the bivariate bicycle codes [[144, 12, 12]]2 and [[90, 8, 10]]2 attain 12 and 8.9). (Lemma 10). Table I collects the best code found at each parameter set; the quantum-LDPC figure of merit kd2/n is used throughout as the quality measure, and for reference the bivariate bicycle code [[1… view at source ↗
Figure 2
Figure 2. Decoding performance: long codes vs BB [[144, 12, 12]]. Markers: Monte Carlo (BP-OSD, 50 iterations, OSD-CS order 1); dashed lines: the fit p˜L(p) = p d/2e α+βp+γp2 on sub-threshold data, drawn over the full axis range for illustration. 10−3 10−2 10−1 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 physical error rate p logical error rate pL [[42,6,7]] [[54,6,8]] [[66,6,8]] [[78,6,8]] [[54,16,6]] [[72,12,6]] [PITH_FULL… view at source ↗
Figure 3
Figure 3. Decoding performance: short codes vs BB [[72, 12, 6]]. Markers: Monte Carlo (BP-OSD, 50 iterations, OSD-CS order 1); dashed lines: the fit p˜L(p) = p d/2e α+βp+γp2 on sub-threshold data, drawn over the full axis range for illustration. the coset generalization of Proposition 4: the first term is the generic (polynomial) contribution, while the rank defect rankHX − rankHZ vanishes identically in the cyclic case and c… view at source ↗

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