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New distance-12 bivariate-bicycle codes with 20 and 24 logical qubits: an exhaustive classification and checkable distances to 360 qubits (certificate release v1.0.0)

Bryan W. Daugherty, Shawn Ryan, Gregory Ward

Published Oct 14, 2026• 0 citations
Publisher

Abstract

We construct three new weight-6 bivariate-bicycle (BB) codes—two [[336,20,12]] codes, inequivalent under the BB presentation symmetries, over Z₂ × Z₈₄ and a [[360,24,12]] code over Z₆ × Z₃₀—and certify exact distance 12 in both the X and Z sectors. They encode 25% and 50% more logical qubits than the best distance-12, weight-6 code reported by IBM's evolutionary search (k = 16) [Cruz-Benito et al., arXiv:2606.02418]. They are also the only ones: a symmetry-reduced enumeration of every connected weight-6 BB code with three-term polynomials, n ≤ 360 and k ≥ 20 eliminates 1,448 of the 1,584 candidates with checked logical operators of weight at most 11, and the remaining 136 presentations fall into three classes under group automorphisms, renormalisation and A ↔ B, represented by the new codes; within the polynomial ansatz of [Bravyi et al., Nature 627, 778 (2024)], every one of the 432 such candidates has distance at most 10. The same certificates settle the distances of the IBM ladder [Bravyi et al., Nature 627, 778 (2024)]—[[72,12,6]], [[90,8,10]], [[108,8,10]], the gross code [[144,12,12]] and the two-gross code [[288,12,18]], both sectors—and certify 19 ≤ d ≤ 24 for [[360,12]], whose distance we establish to be exactly 24 by exhaustive enumeration, resolving the upper bound reported in [Bravyi et al., Nature 627, 778 (2024)]. For the depth-8 syndrome-extraction circuit they certify circuit-level distance exactly 6 for [[72,12,6]] in both memory bases and at least 6 for every other code tested. Certified lower bounds are DRAT refutations for n ≤ 144 and replayable search trees above, whose single pruning lemma is proved in Lean 4; a reader replays any certificate with a standard-library Python verifier plus drat-trim or a 200-line C replayer, without the software that produced it. In a 1.1-million-shot circuit-level benchmark under one frozen BP-OSD decoder, the new codes' block failure probabilities at p = 0.004 are 3.6–6.2 times lower than the gross code's and 3.4–5.6 times higher than the two-gross code's, in both memory bases. Every computation—search, certification, replay and benchmark—ran on one laptop (Apple M3 Pro, 18 GB) with no GPU and no cluster. Contents of the release paper/ — the manuscript (LaTeX source, bibliography and PDF). certificates/ — one manifest per code and sector (<code_id>_<sector>.json), its standard-library Python verifier (verify_qec_<code_id>_<sector>.py), the proofs in proofs/ (DRAT proofs and gzipped proof trees), the problem files and C source of the tree replayer, the Lean 4 minimality lemma (qec_minimality.lean), the weight-6 envelope witnesses with their checkers, the distance table, FORMAT.md (the certificate format) and REPLAY.md (a clean replay of every certificate: 44 of 44 pass). simulation/ — the projected detector error models, the BP-OSD and BP-LSD evaluation scripts, every 2,000-shot chunk record with its seed, the decoder-tuning grid and the joint-model controls. SHA256SUMS — the SHA-256 of every file. Verification In certificates/: python3 verify_qec_<code_id>_<sector>.py rebuilds the code from its polynomials, checks the logical basis by rank, regenerates every CNF or problem file byte for byte and replays the proof; its last line is the verdict. DRAT rows need drat-trim (set DRAT_TRIM); tree rows need a C compiler. The Lean lemma checks with lake env lean qec_minimality.lean in a project with the pinned Lean 4 v4.33.1 and Mathlib 0df444a3 (exit status 0). Integrity: shasum -a 256 -c SHA256SUMS. Hardware Every computation — search, certificate production, replay and the benchmark — ran on one MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory, macOS 15.5), with no GPU and no cluster. License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). The certificate producer is proprietary and not distributed; verification does not need it.

DOI
10.5281/zenodo.23382174

Related papers

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

quant-pharXiv:2609.06572•Sep 8, 2026•0 citations

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.

quant-pharXiv:2609.39395•Oct 1, 2026•0 citations

Orbit reduction for quantum code distance: a counterexample, certified bounds for bivariate bicycle codes, and two published bounds tightened

Jules Falcone

Computing the minimum distance of a quantum LDPC code is easy in one direction and expensive in the other: exhibiting a light logical operator is a search, while proving that no lighter one exists is what actually decides the parameter. This note proves an orbit reduction for the second half. If a group of code automorphisms acts on the qubits with r orbits, the minimum distance is the smallest of r restricted problems, one per orbit representative, so the number of solver calls is the number of orbits and not one. We give the a priori condition under which restricting to a single qubit is legitimate, namely transitivity on all qubits, and a counterexample showing what happens without it: on a hypergraph product code with n = 27 and d = 3, the widely used tool dist-m4ri prints a pair of equal bounds, which its own documentation calls a confirmed exact distance, and that value is 6. The reduction is coupled to a proof producing pipeline in which every nonexistence claim is re-verified by an independent checker, drat-trim or, for the largest closure, cake_lpr, whose soundness is proved inside CakeML. This closes with replayable certificates the distances of the five smaller bivariate bicycle codes of Bravyi et al., up to and including the code with n = 288 and d = 18, whose exactness was published with no replayable artifact and is established here at both ends. For the two larger codes it yields the first lower bounds a third party can re-check from the deposited files: d at least 18 for the code with n = 360, against a published upper bound of 24 and against d at least 11 for the best previously re-executable method, and a first lower bound of any kind for the code with n = 756. A separate and much cheaper ingredient, a sweep over logical operators supported on unions of cosets of a subgroup, tightens two upper bounds printed in a recent table of two block group algebra codes. The code printed there as [[140, 8, at most 16]] carries a logical operator of weight 14; the code printed as [[144, 10, at most 16]] has distance exactly 12, established at both ends and so closed four below the published bound. Both witnesses are re-checked outside any solver by two rank computations over the field with two elements, so neither of these two upper bounds rests on a solver at all. Both entries are printed with the inequality sign in that reference, which states no method for the tables they sit in; nothing it proves is contradicted here, and no error of its authors is corrected. What is improved are two bounds they offered as bounds. Finally we show that the two instance form of the reduction is a consequence of commutativity and not of the two block structure. On two block group algebra codes over non-abelian groups, taken from the published catalogue, the translation subgroup has two orbits for one code and twelve for another, so the number of solver calls has to be computed from the group before any search rather than assumed. On three such codes we certify the minimum distance. The argument that makes one orientation enough in the abelian case does not apply there, and we check that it does not, so both orientations are certified separately rather than assumed equal; on these three codes the two turn out to coincide, which is a measured fact about each code and not a consequence of that argument. The published values for these codes come from randomised search, and a certificate of nonexistence a reader can replay is, to our knowledge, new for non-abelian codes of this family. What this work does not claim is stated explicitly in the note: the restriction itself is not our idea, it is an option of dist-m4ri present since 2020; a machine checkable chain covering a symmetry breaking step is not a first; our proofs are checked by an independent checker and not by the kernel of a proof assistant; and the encoding step is not formally verified. A public question asking whether the orbit reduction is already in the literature was posted on MathOverflow on 28 August 2026 and carried no answer at the time of deposit. Every table of numbers in the note is generated from the deposited artifacts by a script that refuses to emit a row it cannot source. 27 pages. Verification scripts and certificate digests included as separate archives. MSC 2020: 81P73, 94B05, 68Q25, 20C05.

•Aug 29, 2026•0 citations

Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits

Mohammad Rowshan

Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a $k/2$-dimensional annihilator subspace and a $k/2$-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.

quant-pharXiv:2609.39121•Oct 1, 2026•0 citations

The gross code's depth-8 syndrome cycle has circuit distance 10

Maximilian Kornstein

Bravyi et al. conjectured that their depth-8 syndrome cycle for the [[144,12,12]] gross code has circuit distance 10. We prove it. In their noise model and for every number of noisy cycles, no set of nine or fewer faults causes an undetected logical error of either Pauli type, and we give explicit ten-fault logical errors of both types. When we released our data on 15 October 2026, the best reported bounds for X-type faults were 9 ≤ d_circ ≤ 12, at 12 rounds; since then, Crocker and Pan have reported the value 10 in both bases, for 12 noisy cycles and for every number of cycles up to 12, respectively. For the same family of circuits we show that the [[90,8,10]] and [[108,8,10]] cycles have circuit distance 8 for every number of cycles, extending values reported for fixed numbers of rounds; for [[72,12,6]] we confirm the known value 6. We also find that, for three of these codes, Google's Tesseract test circuits have the same detector-error-model columns as the depth-8 cycle once restricted to the detectors of the memory basis, for every number of rounds we checked, so circuit-distance results carry over between the two at those round counts. The lower bounds combine 607 SAT refutations, each with a DRAT proof, with exhaustive enumerations of extended-code logicals that carry no certificate. Finally, we independently confirm two results that others reported first: the [[360,12]] bivariate bicycle code has distance 24, and the left-right gross-code circuit of Strikis, Browne and Beverland, as reconstructed from their figures, has circuit distance 11. All code, formulas, proofs and fault sets are available at doi:10.5281/zenodo.23394575.

•Oct 29, 2026•0 citations

A 16-fault undetectable logical error in the depth-8 syndrome cycle of the two-gross code [[288,12,18]]

Carson Crocker

For the bivariate bicycle code [[288,12,18]] ("two-gross"), with the depth-8 syndrome-extraction cycle of Bravyi et al. (arXiv:2308.07915), this deposit gives explicit sets of 16 circuit faults that trigger no detector and apply a nontrivial logical operator. They exist for Z-type and X-type faults and under both readings of the CNOT term labels. One of them lies entirely inside the noise model and memory layout of Beverland et al., "Fail fast" (arXiv:2511.15177: Z-type noise, 18 noisy cycles and one fault-free cycle), whose search reported a lowest weight of 18. So the circuit distance of this cycle is at most 16. No lower bound is claimed, and no claim is made about logical error rates. Contents. A two-page note; the four fault sets listed gate by gate (cycle, timestep, gate, qubit coordinates, Pauli) so they can be checked by hand; a standalone checker (Python 3 + numpy) that builds the code and cycle from the polynomials and replays each set; tamper controls; logs. Checks. Two separately written circuit builds (one written from the papers only) reproduce the matrix sizes in Fail fast, Table 1, and agree on all four sets at cycles 1, 9 and 18; the gate order and indexing were checked against the text of the public BivariateBicycleCodes repository (inspected, not executed). A third propagator, written from the fault table and the papers only, is included. Removing any single fault makes detectors fire in every case tested. Use of AI. The author is not a professional researcher. The search, checks and drafting were carried out by AI models (Anthropic's Claude via Claude Code; OpenAI's ChatGPT as a reviewer) under the author's direction. The author is responsible for the content. No human expert has reviewed the work to date. Corrections are welcome.

•Oct 21, 2026•0 citations
We construct three new weight-6 bivariate-bicycle (BB) codes—two [[336,20,12]] codes, inequivalent under the BB presentation symmetries, over Z₂ × Z₈₄ and a [[360,24,12]] code over Z₆ × Z₃₀—and certify exact distance 12 in both the X and Z sectors. They encode 25% and 50% more logical qubits than the best distance-12, weight-6 code reported by IBM's evolutionary search (k = 16) [Cruz-Benito et al., arXiv:2606.02418]. They are also the only ones: a symmetry-reduced enumeration of every connected weight-6 BB code with three-term polynomials, n ≤ 360 and k ≥ 20 eliminates 1,448 of the 1,584 candidates with checked logical operators of weight at most 11, and the remaining 136 presentations fall into three classes under group automorphisms, renormalisation and A ↔ B, represented by the new codes; within the polynomial ansatz of [Bravyi et al., Nature 627, 778 (2024)], every one of the 432 such candidates has distance at most 10. The same certificates settle the distances of the IBM ladder [Bravyi et al., Nature 627, 778 (2024)]—[[72,12,6]], [[90,8,10]], [[108,8,10]], the gross code [[144,12,12]] and the two-gross code [[288,12,18]], both sectors—and certify 19 ≤ d ≤ 24 for [[360,12]], whose distance we establish to be exactly 24 by exhaustive enumeration, resolving the upper bound reported in [Bravyi et al., Nature 627, 778 (2024)]. For the depth-8 syndrome-extraction circuit they certify circuit-level distance exactly 6 for [[72,12,6]] in both memory bases and at least 6 for every other code tested. Certified lower bounds are DRAT refutations for n ≤ 144 and replayable search trees above, whose single pruning lemma is proved in Lean 4; a reader replays any certificate with a standard-library Python verifier plus drat-trim or a 200-line C replayer, without the software that produced it. In a 1.1-million-shot circuit-level benchmark under one frozen BP-OSD decoder, the new codes' block failure probabilities at p = 0.004 are 3.6–6.2 times lower than the gross code's and 3.4–5.6 times higher than the two-gross code's, in both memory bases. Every computation—search, certification, replay and benchmark—ran on one laptop (Apple M3 Pro, 18 GB) with no GPU and no cluster. Contents of the release paper/ — the manuscript (LaTeX source, bibliography and PDF). certificates/ — one manifest per code and sector (<code_id><sector>.json), its standard-library Python verifier (verify_qec<code_id><sector>.py), the proofs in proofs/ (DRAT proofs and gzipped proof trees), the problem files and C source of the tree replayer, the Lean 4 minimality lemma (qec_minimality.lean), the weight-6 envelope witnesses with their checkers, the distance table, FORMAT.md (the certificate format) and REPLAY.md (a clean replay of every certificate: 44 of 44 pass). simulation/ — the projected detector error models, the BP-OSD and BP-LSD evaluation scripts, every 2,000-shot chunk record with its seed, the decoder-tuning grid and the joint-model controls. SHA256SUMS — the SHA-256 of every file. Verification In certificates/: python3 verify_qec<code_id>_<sector>.py rebuilds the code from its polynomials, checks the logical basis by rank, regenerates every CNF or problem file byte for byte and replays the proof; its last line is the verdict. DRAT rows need drat-trim (set DRAT_TRIM); tree rows need a C compiler. The Lean lemma checks with lake env lean qec_minimality.lean in a project with the pinned Lean 4 v4.33.1 and Mathlib 0df444a3 (exit status 0). Integrity: shasum -a 256 -c SHA256SUMS. Hardware Every computation — search, certificate production, replay and the benchmark — ran on one MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory, macOS 15.5), with no GPU and no cluster. License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). The certificate producer is proprietary and not distributed; verification does not need it.
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-666 checks, exemplified by [[144,12,12]][[144,12,12]][[144,12,12]] with kd2/n=12kd^2/n=12kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-888 checks (weight-444 generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim⁡R/(A,B)k=2\dim R/(A,B)k=2dimR/(A,B) (forcing even kkk), a 4ℓm4\ell m4ℓm-element symmetry group on generator pairs, an X/ZX/ZX/Z distance equality dX=dZd_X=d_ZdX​=dZ​, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144n=144n=144 the pipeline returns a census of 535353 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]][[144,6,d\ge 15]][[144,6,d≥15]] exceeds the benchmark distance 121212 (certified d≥15d\ge 15d≥15), [[144,10,12]][[144,10,12]][[144,10,12]] reaches it with weight-888 checks, and [[144,16,10]][[144,16,10]][[144,16,10]] encodes a third more logical qubits at kd2/n=11.11kd^2/n=11.11kd2/n=11.11 (7.4%7.4\%7.4% below benchmark) while decoding no worse. At n=72n=72n=72, [[72,14,8]][[72,14,8]][[72,14,8]] attains kd2/n=12.44kd^2/n=12.44kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-888 codes at ≈0.1%\approx 0.1\%≈0.1% pseudo-threshold versus ≈0.4%\approx 0.4\%≈0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the ZZZ-logical quotient K/SK/SK/S over F2[x,y]/(xℓ−1,ym−1)\mathbb F_2[x,y]/(x^\ell-1,y^m-1)F2​[x,y]/(xℓ−1,ym−1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity dZ=min⁡(dA,dC)d_Z=\min(d_{\mathcal A},d_{\mathcal C})dZ​=min(dA​,dC​). Using the Frobenius structure of the finite group algebra, we prove rA=rC=k/2r_{\mathcal A}=r_{\mathcal C}=k/2rA​=rC​=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances 4,6,10,10,12,184,6,10,10,12,184,6,10,10,12,18 of the six standard BB codes of lengths 181818 to 288288288 and enumerates every minimum-weight logical operator. The resulting census shows that [[108,8,10]][[108,8,10]][[108,8,10]] is the only one of the six whose distance is attained in a single component, with dC=10d_{\mathcal C}=10dC​=10 and dA=12d_{\mathcal A}=12dA​=12.
Computing the minimum distance of a quantum LDPC code is easy in one direction and expensive in the other: exhibiting a light logical operator is a search, while proving that no lighter one exists is what actually decides the parameter. This note proves an orbit reduction for the second half. If a group of code automorphisms acts on the qubits with r orbits, the minimum distance is the smallest of r restricted problems, one per orbit representative, so the number of solver calls is the number of orbits and not one. We give the a priori condition under which restricting to a single qubit is legitimate, namely transitivity on all qubits, and a counterexample showing what happens without it: on a hypergraph product code with n = 27 and d = 3, the widely used tool dist-m4ri prints a pair of equal bounds, which its own documentation calls a confirmed exact distance, and that value is 6. The reduction is coupled to a proof producing pipeline in which every nonexistence claim is re-verified by an independent checker, drat-trim or, for the largest closure, cake_lpr, whose soundness is proved inside CakeML. This closes with replayable certificates the distances of the five smaller bivariate bicycle codes of Bravyi et al., up to and including the code with n = 288 and d = 18, whose exactness was published with no replayable artifact and is established here at both ends. For the two larger codes it yields the first lower bounds a third party can re-check from the deposited files: d at least 18 for the code with n = 360, against a published upper bound of 24 and against d at least 11 for the best previously re-executable method, and a first lower bound of any kind for the code with n = 756. A separate and much cheaper ingredient, a sweep over logical operators supported on unions of cosets of a subgroup, tightens two upper bounds printed in a recent table of two block group algebra codes. The code printed there as [[140, 8, at most 16]] carries a logical operator of weight 14; the code printed as [[144, 10, at most 16]] has distance exactly 12, established at both ends and so closed four below the published bound. Both witnesses are re-checked outside any solver by two rank computations over the field with two elements, so neither of these two upper bounds rests on a solver at all. Both entries are printed with the inequality sign in that reference, which states no method for the tables they sit in; nothing it proves is contradicted here, and no error of its authors is corrected. What is improved are two bounds they offered as bounds. Finally we show that the two instance form of the reduction is a consequence of commutativity and not of the two block structure. On two block group algebra codes over non-abelian groups, taken from the published catalogue, the translation subgroup has two orbits for one code and twelve for another, so the number of solver calls has to be computed from the group before any search rather than assumed. On three such codes we certify the minimum distance. The argument that makes one orientation enough in the abelian case does not apply there, and we check that it does not, so both orientations are certified separately rather than assumed equal; on these three codes the two turn out to coincide, which is a measured fact about each code and not a consequence of that argument. The published values for these codes come from randomised search, and a certificate of nonexistence a reader can replay is, to our knowledge, new for non-abelian codes of this family. What this work does not claim is stated explicitly in the note: the restriction itself is not our idea, it is an option of dist-m4ri present since 2020; a machine checkable chain covering a symmetry breaking step is not a first; our proofs are checked by an independent checker and not by the kernel of a proof assistant; and the encoding step is not formally verified. A public question asking whether the orbit reduction is already in the literature was posted on MathOverflow on 28 August 2026 and carried no answer at the time of deposit. Every table of numbers in the note is generated from the deposited artifacts by a script that refuses to emit a row it cannot source. 27 pages. Verification scripts and certificate digests included as separate archives. MSC 2020: 81P73, 94B05, 68Q25, 20C05.
Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a k/2k/2k/2-dimensional annihilator subspace and a k/2k/2k/2-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.
Bravyi et al. conjectured that their depth-8 syndrome cycle for the [[144,12,12]] gross code has circuit distance 10. We prove it. In their noise model and for every number of noisy cycles, no set of nine or fewer faults causes an undetected logical error of either Pauli type, and we give explicit ten-fault logical errors of both types. When we released our data on 15 October 2026, the best reported bounds for X-type faults were 9 ≤ d_circ ≤ 12, at 12 rounds; since then, Crocker and Pan have reported the value 10 in both bases, for 12 noisy cycles and for every number of cycles up to 12, respectively. For the same family of circuits we show that the [[90,8,10]] and [[108,8,10]] cycles have circuit distance 8 for every number of cycles, extending values reported for fixed numbers of rounds; for [[72,12,6]] we confirm the known value 6. We also find that, for three of these codes, Google's Tesseract test circuits have the same detector-error-model columns as the depth-8 cycle once restricted to the detectors of the memory basis, for every number of rounds we checked, so circuit-distance results carry over between the two at those round counts. The lower bounds combine 607 SAT refutations, each with a DRAT proof, with exhaustive enumerations of extended-code logicals that carry no certificate. Finally, we independently confirm two results that others reported first: the [[360,12]] bivariate bicycle code has distance 24, and the left-right gross-code circuit of Strikis, Browne and Beverland, as reconstructed from their figures, has circuit distance 11. All code, formulas, proofs and fault sets are available at doi:10.5281/zenodo.23394575.
For the bivariate bicycle code [[288,12,18]] ("two-gross"), with the depth-8 syndrome-extraction cycle of Bravyi et al. (arXiv:2308.07915), this deposit gives explicit sets of 16 circuit faults that trigger no detector and apply a nontrivial logical operator. They exist for Z-type and X-type faults and under both readings of the CNOT term labels. One of them lies entirely inside the noise model and memory layout of Beverland et al., "Fail fast" (arXiv:2511.15177: Z-type noise, 18 noisy cycles and one fault-free cycle), whose search reported a lowest weight of 18. So the circuit distance of this cycle is at most 16. No lower bound is claimed, and no claim is made about logical error rates. Contents. A two-page note; the four fault sets listed gate by gate (cycle, timestep, gate, qubit coordinates, Pauli) so they can be checked by hand; a standalone checker (Python 3 + numpy) that builds the code and cycle from the polynomials and replays each set; tamper controls; logs. Checks. Two separately written circuit builds (one written from the papers only) reproduce the matrix sizes in Fail fast, Table 1, and agree on all four sets at cycles 1, 9 and 18; the gate order and indexing were checked against the text of the public BivariateBicycleCodes repository (inspected, not executed). A third propagator, written from the fault table and the papers only, is included. Removing any single fault makes detectors fire in every case tested. Use of AI. The author is not a professional researcher. The search, checks and drafting were carried out by AI models (Anthropic's Claude via Claude Code; OpenAI's ChatGPT as a reviewer) under the author's direction. The author is responsible for the content. No human expert has reviewed the work to date. Corrections are welcome.