[[39,17,3]] · QEC Challenge

[[39,17,3]] d =

kd²/n

3.92308

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 9, w_Z = 9 (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)

[4, 19, 32]

d_Z 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)

[2, 6, 35]

certificate exact, d = 3 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 3 exists; Z: no logical < 3 exists

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 · H_Z 9

qubit degrees H_X 1–5 (mean 2.538) · H_Z 1–5 (mean 2.538)

trapping sets H_X (1,1)×8 (2,1)×19 (3,0)×11 (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): 11 (1,3): 13 (1,4): 5 (1,5): 2 (2,1): 19 (2,2): 55 (2,3): 100 (2,4): 92 (2,5): 47 (2,6): 20 (2,7): 4 (3,0): 11 (3,1): 91 (3,2): 358 (3,3): 713 (3,4): 929 (3,5): 835 (3,6): 508 (3,7): 181 (3,8): 57 (3,9): 9 (3,10): 1

trapping sets H_Z (1,1)×8 (2,1)×17 (3,0)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 12 (1,3): 10 (1,4): 8 (1,5): 1 (2,1): 17 (2,2): 56 (2,3): 105 (2,4): 82 (2,5): 55 (2,6): 21 (2,7): 2 (3,0): 6 (3,1): 112 (3,2): 320 (3,3): 730 (3,4): 952 (3,5): 846 (3,6): 485 (3,7): 210 (3,8): 41 (3,9): 8 (3,10): 3

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PUNCTURE transform of codes/40-16-4.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

[[39,17,3]] — priced-calculus transform of 40-16-4.json

Direction & hypothesis

Weight-9plus unrestricted CSS cell. Transform a connected CSS parent (40-16-4.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/40-16-4.json. Rule used for this survivor: E-PUNCTURE. 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/40-16-4.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/40-16-4.json by applying the E-PUNCTURE move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 11 (max weight 9) · Z-checks 11 (max weight 9)

H_X (11 checks, sparse supports)

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

H_Z (11 checks, sparse supports)

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

Code ID 39-17-3 · download JSON · raw on GitHub