[[37,22,3]] · QEC Challenge

[[37,22,3]] d ≤

kd²/n

5.351

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 20, w_Z = 20 (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)

[14, 21, 35]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 11, 25]

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 16–20 (mean 18.5) · H_Z 15–20 (mean 17.714)

qubit degrees H_X 1–7 (mean 4.0) · H_Z 1–6 (mean 3.351)

trapping sets H_X (1,1)×3 (2,1)×24 (3,0)×55 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 3 (1,2): 5 (1,3): 6 (1,4): 7 (1,5): 8 (1,6): 7 (1,7): 1 (2,1): 24 (2,2): 64 (2,3): 121 (2,4): 161 (2,5): 136 (2,6): 55 (2,7): 15 (3,0): 55 (3,1): 268 (3,2): 740 (3,3): 1562 (3,4): 1987 (3,5): 1560 (3,6): 726 (3,7): 168 (3,8): 19

trapping sets H_Z (1,1)×4 (2,1)×27 (3,0)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 6 (1,3): 9 (1,4): 10 (1,5): 7 (1,6): 1 (2,1): 27 (2,2): 103 (2,3): 179 (2,4): 157 (2,5): 76 (2,6): 14 (3,0): 72 (3,1): 410 (3,2): 1185 (3,3): 1966 (3,4): 1867 (3,5): 994 (3,6): 323 (3,7): 60

Circuit tier

syndrome-extraction memory circuits committed under circuits/37-22-3/ · 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)

[1061, 1062]

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

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

[353, 354]

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 37-21-3.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 > 8 (computed)

How this code was found

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

[[37,22,3]] C-DELETE-CHECK b1 from 37-21-3.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=[5, 21, 23], Z=[2, 18, 29]. 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 8 (max weight 20) · Z-checks 7 (max weight 20)

H_X (8 checks, sparse supports)

[0, 1, 3, 5, 7, 8, 10, 12, 14, 15, 16, 17, 19, 22, 23, 25, 27, 35, 36] [0, 1, 2, 4, 5, 6, 7, 9, 11, 13, 14, 15, 16, 18, 20, 21, 22, 24, 35, 36] [1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 16, 18, 20, 23, 27, 33, 34, 36] [1, 2, 3, 5, 7, 8, 12, 14, 16, 18, 19, 21, 32, 33, 34, 36] [0, 1, 2, 4, 5, 9, 10, 11, 13, 15, 16, 19, 29, 30, 31, 33, 35, 36] [0, 3, 4, 8, 9, 10, 11, 12, 14, 15, 18, 28, 29, 30, 31, 32, 34, 35, 36] [1, 2, 3, 7, 8, 9, 10, 13, 14, 17, 27, 28, 29, 30, 33, 34, 35, 36] [0, 2, 6, 7, 8, 9, 11, 12, 13, 16, 26, 27, 28, 29, 31, 32, 33, 34, 35]

H_Z (7 checks, sparse supports)

[0, 3, 5, 10, 11, 13, 14, 15, 16, 20, 22, 23, 24, 25, 26, 29, 34, 35, 36] [1, 2, 7, 8, 10, 12, 14, 16, 17, 20, 30, 33, 34, 35, 36] [0, 1, 2, 3, 5, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 25, 31, 32, 34, 36] [0, 1, 3, 5, 6, 10, 12, 13, 14, 15, 21, 22, 28, 29, 31, 32, 33, 34, 36] [0, 5, 7, 8, 10, 15, 18, 21, 25, 26, 27, 28, 30, 31, 35, 36] [0, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 21, 26, 27, 28, 29, 30, 35] [1, 2, 3, 5, 7, 9, 13, 16, 21, 24, 25, 27, 28, 31, 34, 35]

Code ID 37-22-3 · download JSON · raw on GitHub