“Bivariate bicycle” is a construction principle, while names such as gross, double-gross, BB5, or BB6 often refer to finite instances or check-weight classes. This page organizes genuine variant axes and prevents parameter claims from leaking across them.
mackay2004sparse introduced sparse quantum bicycle codes built from circulant structure. panteleev2021degenerate developed generalized bicycle codes, finite-length searches, and BP-OSD. Modern BB codes replace a single cyclic direction by the two-dimensional abelian group \(\mathbb Z_l\times\mathbb Z_m\).
The exact parent is abelian two-block group-algebra code: wang2023abelian and lin2023twoblock. “Generalized bicycle,” “2BGA,” “lifted product,” and “bivariate bicycle” overlap historically and algebraically, but they are not interchangeable type names.
| axis | ordinary shipped BB | variant | what must be recomputed |
|---|---|---|---|
| number of variables | \(x,y\) | multivariate (voss2025multivariate) | matrix size, locality dimension, rank, distance |
| polynomial restrictions | unconstrained sparse search | coprime subclass (wang2026coprime) | attainable rate, distance, layout |
| boundary | periodic torus | pruned/open or tile boundary (eberhardt2024pruning, liang2025planar, steffan2025tile) | checks, qubits, rank, logical strings, distance |
| duality | ordinary swapped-transpose CSS | self-dual twisted torus (liang2025selfdual) | check weight, Clifford action, finite parameters |
| alphabet | qubits over \(\mathbb F_2\) | prime-dimensional qudits (liang2026qudit) | commutation form, \(n,k,d\) units, decoder |
| CSS condition | exact \(H_X,H_Z\) split | perturbed non-CSS candidates (cruzbenito2026evolutionary) | full symplectic checks and Pauli distance |
| extraction | static code throughout | morphing (shaw2025morphing) | instantaneous stabilizers, detectors, circuit distance |
Adding a third independent cyclic variable produces a multivariate bicycle code. It remains an abelian group-algebra relative, but it is no longer bivariate. The extra direction can change locality dimension and hardware embedding even if the number of polynomial terms stays fixed.
Consequently, a multivariate candidate with the same check weight is not a BB instance with a longer parameter list. The constructor and controlled relation should identify its larger algebra.
wang2026coprime constrains generator factors so rate can be selected before the full finite search and co-designs a cold-atom layout. This is a subclass: every coprime BB candidate satisfies the broader bivariate form, but ordinary BB polynomials need not satisfy the coprime factorization.
The restriction can improve design predictability while excluding otherwise useful finite blocks. Rate formulas from the subclass must not be assigned to the gross code without checking its factors and periods.
A periodic BB code identifies opposite torus edges. Removing qubits/checks or terminating a repeated tile breaks those identifications. Three separate questions appear:
eberhardt2024pruning guarantees its procedure for the cyclic-HGP subcase and demonstrates examples beyond it; that is not a theorem for arbitrary BB polynomials. liang2025planar and steffan2025tile build planar/open codes related to BB bulk patterns, but a shared tile does not determine a shared logical space.
The safe workflow is to construct the finite open matrices, recompute rank, validate commutation, and find new logical representatives. Copying periodic \(k,d\) across the cut is invalid.
wang2026bbs further shows why polynomial roots at zero or infinity require special boundary generators and distinguishes rectangular, diagonal, and tilted prescriptions. “Open BB” is therefore not one boundary condition.
Ordinary BB already has an X/Z-related swapped-transpose form, but that does not automatically give a physical self-duality implementing H or S. liang2025selfdual imposes additional constraints on a twisted torus and reports weight-eight examples with transversal Clifford gates.
Two implications must stay separate:
The first is generic to the shipped construction. The second belongs only to the cited subfamily.
liang2026qudit extends twisted-torus ideas to prime-dimensional qudits. A qudit stabilizer uses a symplectic form over \(\mathbb F_p\), not binary XOR. Its \(k\) counts logical \(p\)-level systems, and its hardware error channel has different Pauli types and normalization.
Comparing \(kd^2/n\) across qubit and qudit examples without stating \(p\) mixes resource units. A future qudit Atlas entry should use an engine type that actually validates \(\mathbb F_p\) commutation rather than forcing the data through the binary constructor.
cruzbenito2026evolutionary combines program mutation, GF(2) rank, distance certification, graph deduplication, and local-Clifford checks. Some candidates leave the strict CSS BB ansatz. Once X and Z are mixed within stabilizer rows, the defining object is a full symplectic check matrix and the CSS polynomial commutation proof no longer applies.
Similarity of a Tanner graph or ancestry in a BB search is not enough to keep the strict family label.
postema2025existence proves asymptotic badness for the studied BB subclass. That result and the strong gross finite point coexist:
The correct product claim is therefore “strong finite constants under a specific connectivity model,” not “asymptotically good qLDPC from a two-dimensional local torus.”
Before transferring a claim between variants, ask:
If any answer is no, the new object needs its own executable instance and provenance ledger.