{"id":"6ead6f34-0ffb-4fce-b75e-ae4cedca8cfc","arxiv_id":"2608.09115","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A polynomial-gcd reformulation of bicycle quantum LDPC codes enables an exact-distance search that finds [[66,20,7]] with kd^2/n=14.85 and proves an n=48 exclusion result.","lead":"The authors turn bicycle quantum error-correcting codes into polynomials, so the number of protected qubits comes from a greatest common divisor before any big matrix is built. Their search finds compact new codes, including one with 20 logical qubits on 66 physical qubits at distance 7, a better figure of merit than the bivariate bicycle code at length 144.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unaudited exact-distance and census computations are the load-bearing risk; Proposition 4's proof has an invalid step but the formula is salvageable.","rationale":"The reader's conditional verdict is appropriate. The most load-bearing assumption is that the exact-distance computations and the exhaustive n=48 census are correct, since the paper's headline parameters and its exclusion theorem cannot be checked by hand from the text. The absence of shipped code, data files, or a reproducible artifact leaves this assumption unverified. The Proposition 4 proof flaw is real and should be fixed, but the theorem itself survives via CRT decomposition of the squarefree ring F2[x]/(x^l-1), so it is not the primary threat to the central claim. As long as the authors release the Magma search, the exact-distance verification, and the census data, or an independent implementation reproduces Example 11 and the seven distance-5 classes, the central claims are likely to stand.","tokens_in":19597,"tokens_out":15161,"duration_ms":164465,"concrete_test":"Independently rerun the reproducibility audit: (1) build H_X=[circ(gu)|circ(gv)] and H_Z=[circ(g*vbar)|circ(g*ubar)] for Example 11, l=33, g={0,3,5,7,10}, u={1,2,3,4,12,13,14,15,16,19,22}, v={0,1,6,8,9,10,11,12,16,19,20,22}, check rank H_X=23, k=20, and d=7 by an independent exact-distance routine that walks kernel weight classes and excludes both stabilizer row spaces; (2) independently re-enumerate all weight-8 symmetric coset-2BGA pairs for G=SmallGroup(72,30), H isomorphic to C3, m=24, and confirm that the seven distance-5 classes all have (rank H_X, rank H_Z)=(19,20), while the six distance-6 classes have full ranks. If either check disagrees, the corresponding central claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claims rest on two computer-assisted computations that the preprint does not make auditable. First, Tables I-V list exact distances obtained by Lemma 10's kernel-weight walk for lengths up to n=200; no Magma code, data files, or commit hash is shipped, and the distance routine is described only as walking weight classes and testing stabilizer membership. Second, Theorem 16 is an exhaustive 123-class census of the weight-8 symmetric coset-2BGA family at n=48; the paper states that proofs are only sketched and that full census data will appear in a companion paper (Section VII). A bug in either computation would invalidate the headline code [[66,20,7]] with kd^2/n=14.85, or the exclusion of [[48,10,5]], respectively. There is also an internal proof flaw in Proposition 4: the claim that Ann_R(a) is contained in Ann_R(b) does not follow from h=gcd(a,b,x^l-1) dividing both b and gcd(a,x^l-1). For example, with l=3, a=x+1 and b=x^2+x+1, Ann(a)=(x^2+x+1) is not contained in Ann(b)=(x+1). The dimension formula k=2 deg gcd(a,b,x^l-1) is nevertheless correct for odd l by CRT decomposition, so this is a proof gap rather than a false theorem; but as printed, the dimension-control proposition is not established by the given argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19933,"tokens_out":9382,"duration_ms":98439,"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":[{"comment":"ok","section":"Section III-C, Proposition 4"},{"comment":"ok","section":"Section IV-B and Tables I-V"},{"comment":"ok","section":"Section VII, Theorem 16"},{"comment":"ok","section":"Section III-A, Lemma 1 and Remark 3"}],"minor_comments":[{"comment":"ok","section":"Section V, paragraph after Table III"},{"comment":"ok","section":"Section V, Table III caption"},{"comment":"ok","section":"Section VI, Table VII"},{"comment":"ok","section":"Section VIII-C"}],"recommendation":"major_revision","confidential_remarks":"The false Lemma 1 and the proof gap in Proposition 4 are technical issues that can be repaired without changing the paper's main thrust. The larger obstacle is computational audibility: both the exact-distance tables and the n=48 census are load-bearing and are not reproducible from the preprint. I would encourage the editor to require release of the Magma code and data files, or an appendix with certificates, before acceptance. I also recommend asking the authors to state clearly that the involution filter in Algorithm 1 is a deliberate restriction of the search space, since Lemma 1 does not show it is automatic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a clean polynomial reformulation of cyclic bicycle codes, and the search it enables finds genuinely good small codes, most notably [[66,20,7]] with kd^2/n=14.85, beating the [[144,12,12]] bivariate bicycle code at less than half the length. The gcd dimension formula is correct for odd l, and the Calderbank-based exact distance computation with the stabilizer excluded is the right way to evaluate these codes. The n=48 results—the minimal 48-element group realization of [[48,10,6]] and the rank-degeneracy theorem excluding [[48,10,5]] from the weight-8 symmetric coset family—are interesting and go beyond the cyclic case.\n\nThe soft spots are real but not fatal. Proposition 4's proof contains a false injectivity assertion: in R=F2[x]/(x^3-1), with a=x+1 and b=x^2+x+1, the annihilator of a is not contained in the annihilator of b. The formula itself holds by CRT decomposition, so this is a proof gap, not a false theorem. The larger concern is auditability: the exact distances in Tables I–V and the exhaustive 123-class census behind Theorem 16 are computer-assisted, but no Magma code, data files, or commit hash are provided. The distance routine is described only sketchily, and the census proofs are explicitly deferred to a companion paper. A bug in either computation would invalidate the headline code or the exclusion claim, so an independent reader currently cannot verify the load-bearing computations. The decoding section fits a three-parameter curve to a modest number of Monte Carlo points and extrapolates to low error rates; the fit parameters are given, but trial counts per point and any error bars would make the comparison firmer.\n\nThe algebraic framework itself is a natural repackaging of known cyclic-code facts, and the paper says so honestly. The genuinely new pieces are the exact-distance pipeline, the specific high-ratio codes, and the rank-degeneracy theorem. The citation pattern is balanced and does not look inflated.\n\nBottom line: this is a solid contribution to finite-length quantum LDPC code search, with a theorem that is true but misproved and computational claims that need to be made auditable. It deserves a serious referee. I would recommend sending it to review, with the conditions that the Proposition 4 proof be replaced by the CRT argument and that the search code and census data be released.","headline":"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.","tokens_in":20468,"tokens_out":2229,"would_cite":true,"duration_ms":25021,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B35","81P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["quantum LDPC codes","bicycle codes","stabilizer codes","cyclic codes","Calderbank correspondence","polynomial gcd","exact distance","coset-2BGA codes"],"falsifier":"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 $\\operatorname{rank} H_X=19$, $\\operatorname{rank} H_Z=20$.","tokens_in":19362,"feed_emoji":"⚛️","tokens_out":18984,"duration_ms":147212,"temperature":0.7,"pith_summary":"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.","feed_headline":"Divisor search finds 66-qubit code with kd^2/n 14.85","feed_subtitle":"The polynomial reformulation turns bicycle code search into a gcd filter and beats the bivariate bicycle benchmark.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the symplectic-to-additive-over-$\\mathbb{F}_4$ correspondence that the exact distance computation relies on, mapping symplectic self-orthogonal codes to additive self-orthogonal codes over $\\mathbb{F}_4$.","marker":"[35]"},{"why":"Defines the bivariate bicycle reference code $[[144,12,12]]_2$, the benchmark for $kd^2/n$ and for the decoding comparisons.","marker":"[11]"},{"why":"Provides the coset-based codes whose $n=48$ parameters ($[[48,10,6]]_2$ and $[[48,8,6]]_2$) are improved or bounded by the paper's constructions and Theorem 16.","marker":"[36]"},{"why":"Introduces generalized bicycle codes, the family the paper reformulates in polynomial form.","marker":"[8]"},{"why":"Introduces two-block group algebra (2BGA) codes, the general setting of which the cyclic case is a specialization.","marker":"[9]"},{"why":"Provides the trivariate bicycle codes whose $kd^2/n$ values the paper's $[[66,20,7]]_2$ exceeds.","marker":"[18]"}],"fun_headline_variants":["Polynomial gcd filter discovers 66-qubit code with kd^2/n 14.85","66-qubit code outperforms bivariate bicycle's kd^2/n","Divisor search finds 66-qubit code with high kd^2/n, beating bivariate","gcd-based search yields 66-qubit LDPC code, beats bivariate benchmark"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial gcd filter discovers 66-qubit code with kd^2/n 14.85","66-qubit code outperforms bivariate bicycle's kd^2/n","Divisor search finds 66-qubit code with high kd^2/n, beating bivariate","gcd-based search yields 66-qubit LDPC code, beats bivariate benchmark"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001221,"raw_usage":{"total_tokens":5257,"prompt_tokens":1414,"completion_tokens":3843,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1030,"completion_tokens_details":{"reasoning_tokens":3748}},"tokens_in":1030,"tokens_out":3843,"duration_ms":25403,"temperature":1.0,"reasoning_tokens":3748,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:14:22.036704+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\operatorname{rank} H_X=19$, $\\operatorname{rank} H_Z=20$.","supporting_citations":[{"cited_title":"Quantum error correction via codes over GF(4),","cited_arxiv_id":null,"evidence_quote":"Supplies the symplectic-to-additive-over-$\\mathbb{F}_4$ correspondence that the exact distance computation relies on, mapping symplectic self-orthogonal codes to additive self-orthogonal codes over $\\mathbb{F}_4$."},{"cited_title":"Quantum kronecker sum-product low-density parity-check codes with finite rate,","cited_arxiv_id":null,"evidence_quote":"Introduces generalized bicycle codes, the family the paper reformulates in polynomial form."},{"cited_title":"Quantum two-block group algebra codes,","cited_arxiv_id":null,"evidence_quote":"Introduces two-block group algebra (2BGA) codes, the general setting of which the cyclic case is a specialization."}],"review_version":1}