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[[12,8,2]] d ≤
n
12
k
8
d
2
kd²/n
2.66667
w
8
X/Z
1
g
0.0631
r
3.6056
layers
1
swaps
5

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[10, 11]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 1]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X acyclic · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–8 (mean 6.0) · H_Z 8
qubit degrees H_X 1 · H_Z 1–2 (mean 1.333)
trapping sets H_X (1,1)×12 (2,0)×34 (3,1)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 12 (2,0): 34 (3,1): 60
trapping sets H_Z (1,1)×8 (2,0)×18 (3,0)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 8 (1,2): 4 (2,0): 18 (2,1): 32 (3,0): 64 (3,1): 56 (3,2): 52
witness diameter X 2.0 · Z 1.0 (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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-09-26
r = 3.606
X checkZ checkqubit site (12)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 5 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction C-ADD-CHECK strict improvement (code calculus v0.2rc): added one check killing all weight-1 logicals of parent codes/12-9-1-b.json
model Beam 1.0 (claimed, not verified)
date 2026-10-24
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent; its interaction radius was recomputed against this entry's own checks.
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[12,8,2]] — C-ADD-CHECK strict improvement of 12-9-1-b

Direction & hypothesis

Convert a d=1 parent (codes/12-9-1-b.json) to d=2 by C-ADD-CHECK: the parent's distance is 1, so its minimum-weight logicals are bare (weight-1) qubits. Adding a single check in the opposite sector whose support contains every bare logical qubit kills them all by anticommutation, raising that sector's distance to at least 2 (Proposition 18.3, code calculus v0.2rc).

What was searched

For the parent, enumerate the weight-1 logicals in the sector whose distance is 1 (O(n) singletons). If their union is an admissible check support (commutes with the sector's own checks, independent in the added sector), build the child and verify witnesses. No exponential search was needed.

Evidence trail

  • Parent: codes/12-9-1-b.json (d=1, upper bound).
  • Child witnesses: inherited from the parent where still valid, plus a

    • weight-2 witness in the converted sector found by pair scan.

  • Verified by verify/validate_candidate.py (passed: true), including the

    • random-seed refutation pass.

Dead ends

Codes where the bare-qubit support was inadmissible (commutation or independence failed) were skipped; some d=1 codes have bare logicals in both sectors and need two checks.

Tools

Model: Beam 1.0. Author: @MathysRennela. Construction calculus v0.2rc (C-ADD-CHECK strict improvement).

Reproduction

Start from codes/12-9-1-b.json, add one check in the opposite sector with support = sorted list of qubits q such that {q} is a weight-1 logical of the parent in the d=1 sector.

Parity checks

X-checks 2 (max weight 8) · Z-checks 2 (max weight 8)
H_X (2 checks, sparse supports)
[0, 1, 4, 5, 6, 7, 10, 11] [2, 3, 8, 9]
H_Z (2 checks, sparse supports)
[0, 1, 2, 3, 6, 7, 8, 9] [2, 3, 4, 5, 8, 9, 10, 11]
Code ID 12-8-2-d · download JSON · raw on GitHub