[[34,17,3]] · QEC Challenge

[[34,17,3]] d ≤

kd²/n

4.5

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)

[1, 25, 31]

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)

[0, 5, 13, 29]

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 2.647) · H_Z 1–4 (mean 2.353)

trapping sets H_X (1,1)×6 (2,1)×8 (3,1)×173 (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): 8 (1,3): 13 (1,4): 6 (1,5): 1 (2,1): 8 (2,2): 66 (2,3): 98 (2,4): 93 (2,5): 52 (2,6): 8 (2,7): 2 (3,1): 173 (3,2): 316 (3,3): 837 (3,4): 1082 (3,5): 587 (3,6): 336 (3,7): 143 (3,8): 12

trapping sets H_Z (1,1)×8 (2,1)×30 (3,0)×29 (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): 11 (1,3): 10 (1,4): 5 (2,1): 30 (2,2): 71 (2,3): 106 (2,4): 65 (2,5): 28 (2,6): 2 (3,0): 29 (3,1): 192 (3,2): 565 (3,3): 884 (3,4): 780 (3,5): 430 (3,6): 166 (3,7): 48

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus C-DELETE-CHECK transform of codes/34-16-4.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

[[34,17,3]] — priced-calculus transform of 34-16-4.json

Direction & hypothesis

Weight-9plus unrestricted CSS cell. Transform a connected CSS parent (34-16-4.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/34-16-4.json. Rule used for this survivor: C-DELETE-CHECK. 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/34-16-4.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/34-16-4.json by applying the C-DELETE-CHECK move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 9 (max weight 10) · Z-checks 8 (max weight 10)

H_X (9 checks, sparse supports)

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

H_Z (8 checks, sparse supports)

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

Code ID 34-17-3-b · download JSON · raw on GitHub