[[14,7,2]] · QEC Challenge

[[14,7,2]] d ≤

kd²/n

2.0

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 8]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 8]

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 4 · H_Z 4 (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 1–3 (mean 1.286) · H_Z 1–3 (mean 1.714)

trapping sets H_X (1,1)×12 (2,0)×19 (3,1)×120 (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 (1,3): 2 (2,0): 19 (2,2): 24 (3,1): 120 (3,3): 36

trapping sets H_Z (1,1)×6 (2,0)×7 (3,1)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 6 (1,2): 6 (1,3): 2 (2,0): 7 (2,1): 12 (2,2): 24 (2,3): 12 (3,1): 108 (3,2): 72 (3,3): 12 (3,4): 32

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a2 from 14-6-2.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

[[14,7,2]] C-DELETE-CHECK a2 from 14-6-2.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=[2, 9], Z=[3, 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 6) · Z-checks 4 (max weight 6)

H_X (3 checks, sparse supports)

[0, 1, 2, 7, 8, 9] [2, 4, 5, 9, 11, 12] [2, 3, 6, 9, 10, 13]

H_Z (4 checks, sparse supports)

[0, 4, 5, 7, 11, 12] [0, 1, 6, 7, 8, 13] [0, 2, 3, 7, 9, 10] [1, 3, 5, 8, 10, 12]

Code ID 14-7-2 · download JSON · raw on GitHub