A bivariate-bicycle (BB) code is defined by two polynomials A, B over F2[x,y]/(x^l - 1, y^m - 1) and has n = 2lm qubits. We fix the polynomials in two families and vary only (l, m) with n <= 400, and report the exact distance of 1027 codes. We then give an upper bound on the distance, built from logicals confined to unions of shifted cosets of a cyclic subgroup H in each block (r <= 2 for every H, and r = 3 when the number of cosets is at most 12), and compare it with the data. It equals the exact distance on 1014 of the 1027 codes (an in-sample fit that is not blind). Two blind tests were run with the predictor and predictions hashed before any distance was read. On 25 distinct catalogued codes it is tight on 24 (96%). On 150 random weight-6 BB codes it is tight on only 102 (68%, 95% confidence interval 0.599-0.754), which fails the preregistered criterion of at least 90%. The bound is never below the exact distance in either test. The bound is therefore tight on 24 of 25 catalogued codes but not on random codes, and the empirical conjecture of Section 5 is restricted to the two families studied.
A bivariate-bicycle (BB) code is defined by two polynomials A, B over F2[x,y]/(x^l - 1, y^m - 1) and has n = 2lm qubits. We fix the polynomials in two families and vary only (l, m) with n <= 400, and report the exact distance of 1027 codes. We then give an upper bound on the distance, built from logicals confined to unions of shifted cosets of a cyclic subgroup H in each block (r <= 2 for every H, and r = 3 when the number of cosets is at most 12), and compare it with the data. It equals the exact distance on 1014 of the 1027 codes (an in-sample fit that is not blind). Two blind tests were run with the predictor and predictions hashed before any distance was read. On 25 distinct catalogued codes it is tight on 24 (96%). On 150 random weight-6 BB codes it is tight on only 102 (68%, 95% confidence interval 0.599-0.754), which fails the preregistered criterion of at least 90%. The bound is never below the exact distance in either test. The bound is therefore tight on 24 of 25 catalogued codes but not on random codes, and the empirical conjecture of Section 5 is restricted to the two families studied.
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 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). Using the Frobenius structure of the finite group algebra, we prove rA=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 4,6,10,10,12,18 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 dC=10 and dA=12.
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 kd2/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=2dimR/(A,B) (forcing even k), a 4ℓm-element symmetry group on generator pairs, an X/Z distance equality dX=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=144 the pipeline returns a census of 53 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]] exceeds the benchmark distance 12 (certified d≥15), [[144,10,12]] reaches it with weight-8 checks, and [[144,16,10]] encodes a third more logical qubits at kd2/n=11.11 (7.4% below benchmark) while decoding no worse. At n=72, [[72,14,8]] attains kd2/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 ≈0.1% pseudo-threshold versus ≈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.
The distance of a quantum code is a fundamental metric that dictates its error-correction capability. While quantum low-density parity-check (qLDPC) families such as Bivariate Bicycle (BB) codes offer a promising pathway to low-overhead fault-tolerant quantum computing, computing their distance is challenging. Current exact solvers and even search heuristics that provide a distance upper bound suffer from severe computational bottlenecks, restricting the discovery of optimal codes. In this paper, we construct a dataset of BB codes with corresponding distance upper bounds obtained through high-budget heuristic searches. We introduce BB-Net, an attention-based neural network architecture for predicting the distance of BB codes. It achieves 96.39% accuracy on the reference distances in the test set, compared with 94.86% for state-of-the-art heuristics with a budget of up to 104 times more computational runtime. On a subset with certified exact distances, BB-Net achieves 98.94%. By relieving the distance-calculation bottleneck, BB-Net enables faster screening of large code spaces, particularly in ML-driven search pipelines, and thereby can accelerate the discovery of novel quantum error-correcting codes, an important capability for advancing scalable fault-tolerant quantum computing. Our code is publicly available: https://github.com/NadavSc/BBNet
We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a h-fold covering of the base BB code's Tanner graph, we refer to it as a h-\textit{cover code}. We show that for a BB code to be a h-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the [[144,12,12]] gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many new codes, including a [[64,14,8]] and [[144,14,14]] code. For an h-cover code of an [[n,k,d]] BB code with parameters [[nh=hn,kh,dh]], we prove that kh≥k for any h, and dh≤hd when h is odd. Furthermore if h is odd and kh=k, we prove the distance lower bound d≤dh. We conjecture it is always true that an h-cover BB code of a base [[n,k,d]] BB code obeys the distance bounds d≤dh≤hd. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.
We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D3, a pair of weighted permutation shifts generates a regular C6×C12 action and recovers the known [[144,12,12]] and [[144,14,14]] bivariate bicycle codes. We also present a computer-verified [[144,16,12]] code over A4×C6, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over F2[C3] yields a [[18,4,3]] code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.
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.