[[12,9,1]] · QEC Challenge

[[12,9,1]] d ≤

kd²/n

0.75

X/Z

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Distance

X/Z asymmetry 2 · d_X ≤ 2, d_Z ≤ 1 · 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)

[3, 9]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[11]

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 8 · H_Z 8

qubit degrees H_X 0–1 (mean 0.667) · H_Z 1–2 (mean 1.333)

trapping sets H_X (1,0)×4 (2,0)×28 (3,1)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,0): 4 (1,1): 8 (2,0): 28 (3,1): 56

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

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a1 from 12-8-2.json

model Beam 1.0 (claimed, not verified)

date 2026-10-19

family other (a tag, not a ranking)

locality unrestricted (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

[[12,9,1]] C-DELETE-CHECK a1 from 12-8-2.json

Direction & hypothesis

Board-advancing code found by the priced construction calculus. Operation: C-DELETE-CHECK applied to a board entry, raising k / lowering check weight while keeping a witness-backed distance. Rule used: C-DELETE-CHECK.

What was searched

A single deletion move on an existing board code, validated by the trusted gate.

Evidence trail

Distance witnesses: X=[10, 11], Z=[4]. The gate's refutation search found no lighter logical.

Dead ends

None recorded for this candidate.

Tools

Model: Beam 1.0. The local verdict is verify/validate_candidate.py.

Reproduction

Reconstruct the parent board code, delete the named check, and recompute n, k, and the witness-backed distance. The resulting checks are the submitted ones.

Parity checks

X-checks 1 (max weight 8) · Z-checks 2 (max weight 8)

H_X (1 checks, sparse supports)

[0, 1, 2, 3, 6, 7, 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 12-9-1 · download JSON · raw on GitHub