[[40,21,3]] · QEC Challenge

[[40,21,3]] d ≤

kd²/n

4.725

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 10, w_Z = 10 (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)

[4, 31, 34]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[11, 12, 30]

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 10 · H_Z 10

qubit degrees H_X 1–5 (mean 2.25) · H_Z 1–5 (mean 2.5)

trapping sets H_X (1,1)×8 (2,1)×45 (3,0)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 19 (1,3): 9 (1,4): 3 (1,5): 1 (2,1): 45 (2,2): 108 (2,3): 105 (2,4): 64 (2,5): 25 (2,6): 3 (3,0): 46 (3,1): 278 (3,2): 732 (3,3): 1008 (3,4): 917 (3,5): 574 (3,6): 191 (3,7): 54 (3,8): 5

trapping sets H_Z (1,1)×4 (2,1)×25 (3,0)×40 (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): 20 (1,3): 9 (1,4): 6 (1,5): 1 (2,1): 25 (2,2): 104 (2,3): 97 (2,4): 101 (2,5): 44 (2,6): 7 (2,7): 2 (3,0): 40 (3,1): 156 (3,2): 626 (3,3): 937 (3,4): 1180 (3,5): 836 (3,6): 381 (3,7): 148 (3,8): 14 (3,9): 2

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a1 from 40-20-3.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

[[40,21,3]] C-DELETE-CHECK a1 from 40-20-3.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=[4, 31, 34], Z=[11, 14, 37]. 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 9 (max weight 10) · Z-checks 10 (max weight 10)

H_X (9 checks, sparse supports)

[1, 2, 16, 18, 21, 23, 24, 25, 28, 31] [1, 2, 17, 19, 20, 22, 26, 27, 28, 31] [1, 3, 4, 7, 24, 26, 28, 29, 33, 34] [0, 2, 5, 6, 25, 27, 28, 29, 32, 35] [5, 7, 8, 11, 28, 30, 32, 33, 37, 38] [4, 6, 9, 10, 29, 31, 32, 33, 36, 39] [9, 11, 12, 15, 21, 22, 32, 34, 36, 37] [8, 10, 13, 14, 20, 23, 33, 35, 36, 37] [13, 15, 16, 19, 20, 21, 25, 26, 36, 38]

H_Z (10 checks, sparse supports)

[0, 2, 13, 14, 16, 17, 20, 23, 25, 27] [1, 3, 12, 15, 16, 17, 21, 22, 24, 26] [0, 2, 12, 15, 18, 19, 21, 22, 25, 27] [0, 1, 4, 6, 17, 18, 24, 27, 29, 31] [0, 1, 5, 7, 16, 19, 25, 26, 28, 30] [1, 2, 4, 5, 8, 10, 28, 31, 33, 35] [0, 3, 4, 5, 9, 11, 29, 30, 32, 34] [5, 6, 8, 9, 12, 14, 32, 35, 37, 39] [4, 7, 8, 9, 13, 15, 33, 34, 36, 38] [9, 10, 12, 13, 16, 18, 21, 23, 36, 39]

Code ID 40-21-3 · download JSON · raw on GitHub