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[[31,15,3]] d =
n
31
k
15
d
3
kd²/n
4.35484
w
8
X/Z
1
g
0.0134
r
4.2426
layers
2
swaps
29

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (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, 21, 27]
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)
[14, 18, 19]
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 8 · H_Z 8
qubit degrees H_X 1–4 (mean 2.065) · H_Z 1–3 (mean 2.065)
trapping sets H_X (1,1)×7 (2,1)×38 (3,0)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 7 (1,2): 16 (1,3): 7 (1,4): 1 (2,1): 38 (2,2): 73 (2,3): 68 (2,4): 24 (2,5): 3 (3,0): 25 (3,1): 187 (3,2): 429 (3,3): 512 (3,4): 332 (3,5): 148 (3,6): 46 (3,7): 6
trapping sets H_Z (1,1)×7 (2,1)×41 (3,0)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 7 (1,2): 15 (1,3): 9 (2,1): 41 (2,2): 75 (2,3): 67 (2,4): 17 (3,0): 31 (3,1): 196 (3,2): 428 (3,3): 474 (3,4): 301 (3,5): 105 (3,6): 24 (3,7): 2
witness diameter X 3.6056 · Z 2.2361 (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)

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/32-14-4.json: the parent's coordinates with the entry of the removed qubit (parent index 13) 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 = 4.243
X checkZ checkqubit site (16)2 qubits stacked (2 layers)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 29 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/32-14-4.json
model Beam 1.0 (claimed, not verified)
date 2026-10-31
family other (a tag, not a ranking)
locality 2D-local bilayer (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

[[31,15,3]] — priced-calculus transform of 32-14-4.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (32-14-4.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/32-14-4.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/32-14-4.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/32-14-4.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 8 (max weight 8) · Z-checks 8 (max weight 8)
H_X (8 checks, sparse supports)
[5, 12, 14, 16, 18, 20, 26, 29] [1, 4, 5, 6, 10, 20, 22, 27] [1, 7, 9, 13, 17, 25, 28, 29] [8, 10, 16, 21, 27, 28, 29, 30] [0, 7, 8, 11, 23, 24, 26, 27] [0, 3, 5, 11, 13, 15, 17, 30] [3, 6, 9, 11, 14, 17, 19, 24] [0, 2, 4, 9, 12, 15, 17, 26]
H_Z (8 checks, sparse supports)
[0, 4, 8, 9, 17, 22, 25, 28] [0, 3, 4, 5, 12, 15, 19, 23] [0, 7, 9, 11, 13, 14, 17, 26] [1, 5, 8, 15, 16, 21, 26, 29] [6, 13, 15, 16, 20, 24, 26, 29] [2, 12, 18, 19, 21, 22, 24, 27] [2, 10, 11, 16, 17, 20, 23, 25] [10, 11, 14, 18, 19, 22, 24, 30]
Code ID 31-15-3 · download JSON · raw on GitHub