X/Z asymmetry 1 · d_X = 10, d_Z = 10 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 10 · witness weight 10 (claimed upper_bound)

d_Z 10 · witness weight 10 (claimed upper_bound)

certificate exact, d = 10 · scipy/HiGHS MILP

X: no logical < 10 exists; Z: no logical < 10 exists

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×84 (2,4)×630 (3,3)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

trapping sets H_Z (1,3)×84 (2,4)×630 (3,3)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

witness diameter X 6.4031 · Z 7.2801 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

layout contributed by @willzeng · Simulated annealing over integer-grid site assignments (spectral initialization, Metropolis moves with softmax cost, max-focused hill-climb polish); 22 independent runs converged to radius 5.0 or 5.099. Search driven by Claude Fable 5. · 2026-08-20

routing cost 335 nearest-neighbor SWAPs per round in total, at most 8 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

authors Zijian Liang and Ke Liu and Hao Song and Yu-An Chen

provenance literature baseline

construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x^-2, g(x,y)=1+y+x^-2y² on the abelian quotient group Z²/L with twist basis a_1=[0, 14], a_2=[3, -6] (n=2|det[a_1,a_2]|=2*42). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].

model classical construction (no AI model)

reference arXiv:2503.03827

date 2025

notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery. LAYOUT added 2026-08-20: a two-layer planar layout with measured interaction radius exactly 5.0 (integer-grid sites, min site spacing 1.0, at most 2 qubits per site), placing the code in the local-2d-bilayer class with kd^2/n = 7.14 (previous weight-6 bilayer cell best 6.00, codes/72-12-6.json) and geometric efficiency g = 0.0114. Layout credit and method are recorded in locality.contributed_by; the CODE, its distance claims, and its authorship are unchanged from the literature baseline. A structured product-fold layout derived from the code's Z_42 translation symmetry bottoms out at sqrt(52) ~ 7.21, so the annealed non-product layout is strictly better and likely at or near this code's floor.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

X-checks 42 (max weight 6) · Z-checks 42 (max weight 6)

Code ID 84-6-10 · download JSON · raw on GitHub