[[39,18,3]] · QEC Challenge

[[39,18,3]] d ≤

kd²/n

4.15385

X/Z

1.33

Share this result

Distance

X/Z asymmetry 1.33 · d_X ≤ 3, d_Z ≤ 4 · 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 found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-10-31

witness operator (support, 3 qubits)

[16, 19, 20]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-10-31

witness operator (support, 4 qubits)

[2, 5, 26, 34]

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 9–10 (mean 9.636) · H_Z 10

qubit degrees H_X 1–5 (mean 2.718) · H_Z 1–4 (mean 2.564)

trapping sets H_X (1,1)×6 (2,1)×9 (3,1)×133 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 6 (1,2): 11 (1,3): 12 (1,4): 8 (1,5): 2 (2,1): 9 (2,2): 60 (2,3): 112 (2,4): 92 (2,5): 65 (2,6): 31 (2,7): 9 (3,1): 133 (3,2): 313 (3,3): 734 (3,4): 1136 (3,5): 1024 (3,6): 643 (3,7): 310 (3,8): 91 (3,9): 16 (3,10): 2

trapping sets H_Z (1,1)×6 (2,1)×13 (3,0)×17 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 6 (1,2): 9 (1,3): 20 (1,4): 4 (2,1): 13 (2,2): 88 (2,3): 110 (2,4): 125 (2,5): 50 (2,6): 2 (3,0): 17 (3,1): 137 (3,2): 501 (3,3): 1060 (3,4): 1220 (3,5): 994 (3,6): 466 (3,7): 110 (3,8): 35

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/40-18-4.json

model Beam 1.0 (claimed, not verified)

date 2026-10-31

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

[[39,18,3]] — priced-calculus transform of 40-18-4.json

Direction & hypothesis

Weight-9plus unrestricted CSS cell. Transform a connected CSS parent (40-18-4.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Single-step deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev @ 71c1adf (machine/priced_calculus.py, machine/planner.py). Parent: codes/40-18-4.json. Rule used for this survivor: E-PEEL. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side (research/kit/surrogate.lightest_logical, seeds 0/1), then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. The gate's random-seed refutation found no lighter logical. Parent entry: codes/40-18-4.json.

Dead ends

Other single-step moves from the same parents were discarded as disconnected, bare-qubit children, dominated candidates, or refuted by the gate.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer. Approx compute: CPU only.

Reproduction

Rebuild H_X/H_Z from codes/40-18-4.json by applying the E-PEEL move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 11 (max weight 10) · Z-checks 10 (max weight 10)

H_X (11 checks, sparse supports)

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

H_Z (10 checks, sparse supports)

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

Code ID 39-18-3 · download JSON · raw on GitHub