[[30,9,3]] · QEC Challenge

[[30,9,3]] d ≤

kd²/n

2.7

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[1, 5, 14]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[7, 13, 27]

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 6 · H_Z 6 (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 1.667) · H_Z 1–2 (mean 1.833)

trapping sets H_X (1,1)×10 (2,1)×40 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 20 (2,1): 40 (2,2): 60 (3,0): 30 (3,1): 120 (3,2): 150 (3,3): 60 (3,4): 40

trapping sets H_Z (1,1)×5 (2,1)×20 (3,0)×20 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 5 (1,2): 25 (2,1): 20 (2,2): 90 (3,0): 20 (3,1): 70 (3,2): 280 (3,3): 30 (3,4): 80

Circuit tier

syndrome-extraction memory circuits committed under circuits/30-9-3/ · canonical noise recipe, 3 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)

[1404, 1415]

d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[1331, 1332, 1416]

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 a7 from 30-8-3.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

[[30,9,3]] C-DELETE-CHECK a7 from 30-8-3.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, 5, 14], Z=[7, 13, 27]. 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 10 (max weight 5) · Z-checks 11 (max weight 5)

H_X (10 checks, sparse supports)

[2, 3, 5, 8, 26] [6, 7, 11, 13, 18] [4, 5, 16, 17, 20] [2, 19, 20, 22, 25] [15, 19, 21, 26, 29] [1, 6, 9, 16, 24] [10, 12, 13, 27, 29] [7, 14, 21, 25, 27] [1, 10, 18, 23, 28] [0, 3, 9, 17, 23]

H_Z (11 checks, sparse supports)

[2, 3, 9, 22, 24] [11, 13, 19, 22, 29] [1, 10, 14, 24, 27] [0, 15, 17, 19, 20] [4, 5, 14, 21, 26] [0, 6, 9, 12, 13] [6, 7, 16, 20, 25] [1, 5, 8, 16, 28] [4, 11, 17, 18, 23] [3, 10, 23, 26, 29] [7, 15, 18, 21, 28]

Code ID 30-9-3 · download JSON · raw on GitHub