[[36,20,3]] · QEC Challenge

[[36,20,3]] d ≤

kd²/n

5.0

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 20, w_Z = 19 (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)

[6, 24, 25]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[3, 10, 22]

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 17–20 (mean 18.375) · H_Z 14–19 (mean 16.75)

qubit degrees H_X 1–6 (mean 4.083) · H_Z 1–6 (mean 3.722)

trapping sets H_X (1,1)×3 (2,1)×23 (3,0)×62 (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): 3 (1,3): 5 (1,4): 9 (1,5): 9 (1,6): 7 (2,1): 23 (2,2): 57 (2,3): 113 (2,4): 156 (2,5): 141 (2,6): 61 (2,7): 7 (3,0): 62 (3,1): 241 (3,2): 703 (3,3): 1390 (3,4): 1811 (3,5): 1558 (3,6): 701 (3,7): 136 (3,8): 5

trapping sets H_Z (1,1)×3 (2,1)×18 (3,0)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 3 (1,2): 5 (1,3): 6 (1,4): 10 (1,5): 9 (1,6): 3 (2,1): 18 (2,2): 52 (2,3): 134 (2,4): 166 (2,5): 110 (2,6): 38 (2,7): 10 (3,0): 25 (3,1): 204 (3,2): 756 (3,3): 1498 (3,4): 1705 (3,5): 1291 (3,6): 634 (3,7): 173 (3,8): 22

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a5 from 36-19-3.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

[[36,20,3]] C-DELETE-CHECK a5 from 36-19-3.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=[2, 20, 29], Z=[16, 18, 22]. 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 8 (max weight 19)

H_X (8 checks, sparse supports)

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

H_Z (8 checks, sparse supports)

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

Code ID 36-20-3 · download JSON · raw on GitHub