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[[15,3,3]] d =
n
15
k
3
d
3
kd²/n
1.8
w
4
X/Z
1
g
0.113
r
2.8284
layers
1
swaps
15

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 4, w_Z = 4 (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)
[2, 6, 13]
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)
[3, 6, 14]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 1–2 (mean 1.6) · H_Z 1–2 (mean 1.6)
trapping sets H_X (1,1)×6 (2,1)×18 (3,0)×9 (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): 9 (2,1): 18 (2,2): 18 (3,0): 9 (3,1): 36 (3,2): 36 (3,3): 18 (3,4): 6
trapping sets H_Z (1,1)×6 (2,1)×18 (3,0)×9 (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 (2,1): 18 (2,2): 18 (3,0): 9 (3,1): 36 (3,2): 36 (3,3): 18 (3,4): 6
witness diameter X 2.0 · Z 2.8284 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/15-3-3/ · canonical noise recipe, 3 rounds, stim 1.16.0
d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[17, 36, 37]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[990, 991, 1407]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @vprusso · Inherited from the laid-out parent codes/16-2-4.json: the parent's coordinates with the entry of the removed qubit (parent index 10) dropped, since this puncture child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 2.828
X checkZ checkqubit site (15)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 15 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PUNCTURE transform of codes/16-2-4.json
model Beam 1.0 (claimed, not verified)
date 2026-10-31
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[15,3,3]] — priced-calculus transform of 16-2-4.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (16-2-4.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/16-2-4.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/16-2-4.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/16-2-4.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 6 (max weight 4) · Z-checks 6 (max weight 4)
H_X (6 checks, sparse supports)
[0, 3, 11, 14] [1, 2, 12, 13] [0, 1, 4, 5] [2, 3, 6, 7] [4, 7, 8, 10] [8, 9, 11, 12]
H_Z (6 checks, sparse supports)
[0, 1, 11, 12] [2, 3, 13, 14] [0, 3, 4, 7] [1, 2, 5, 6] [4, 5, 8, 9] [8, 10, 11, 14]
Code ID 15-3-3 · download JSON · raw on GitHub