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[[12,6,2]] d =
n
12
k
6
d
2
kd²/n
2.0
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 5, w_Z = 5 (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)
[1, 6]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[8, 10]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 1–2 (mean 1.25) · H_Z 1–2 (mean 1.25)
trapping sets H_X (1,1)×9 (2,0)×15 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 9 (1,2): 3 (2,0): 15 (2,1): 12 (3,0): 12 (3,1): 22 (3,2): 7
trapping sets H_Z (1,1)×9 (2,0)×15 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 9 (1,2): 3 (2,0): 15 (2,1): 12 (3,0): 12 (3,1): 22 (3,2): 7

Circuit tier

syndrome-extraction memory circuits committed under circuits/12-6-2/ · canonical noise recipe, 2 rounds, stim 1.16.0
d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[36, 37]
d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[15, 16]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction C-DELETE-CHECK b1 from 12-5-2.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 ≤ 6 (computed)

How this code was found

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

[[12,6,2]] C-DELETE-CHECK b1 from 12-5-2.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=[1, 3], Z=[8, 10]. 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 3 (max weight 5) · Z-checks 3 (max weight 5)
H_X (3 checks, sparse supports)
[0, 1, 7, 9, 11] [2, 3, 6, 8, 10] [4, 5, 6, 8, 10]
H_Z (3 checks, sparse supports)
[1, 3, 5, 6, 7] [0, 2, 4, 8, 9] [0, 2, 4, 10, 11]
Code ID 12-6-2 · download JSON · raw on GitHub