[[28,7,2]] · QEC Challenge

[[28,7,2]] d ≤

kd²/n

1.0

X/Z

Share this result

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)

[24, 25]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[0, 23]

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 5 · H_Z 5

qubit degrees H_X 1–3 (mean 1.786) · H_Z 1–3 (mean 1.964)

trapping sets H_X (1,1)×11 (2,0)×9 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 11 (1,2): 12 (1,3): 5 (2,0): 9 (2,1): 15 (2,2): 36 (2,3): 31 (2,4): 6 (3,0): 4 (3,1): 49 (3,2): 158 (3,3): 147 (3,4): 60 (3,5): 22 (3,6): 12

trapping sets H_Z (1,1)×9 (2,0)×7 (3,1)×33 (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): 11 (1,3): 8 (2,0): 7 (2,1): 9 (2,2): 36 (2,3): 33 (2,4): 21 (3,1): 33 (3,2): 114 (3,3): 220 (3,4): 130 (3,5): 24 (3,6): 24 (3,7): 7

Circuit tier

syndrome-extraction memory circuits committed under circuits/28-7-2/ · 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)

[31, 32]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[137, 138]

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 a1 from 28-6-2.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

[[28,7,2]] C-DELETE-CHECK a1 from 28-6-2.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=[18, 19], Z=[8, 9]. 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)

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

H_Z (11 checks, sparse supports)

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

Code ID 28-7-2 · download JSON · raw on GitHub