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[[54,36,3]] d =
n
54
k
36
d
3
kd²/n
6.0
w
27
X/Z
1
g
0.0037
r
6.3246
layers
2
swaps
32

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 27, w_Z = 27 (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 · 2026-11-01
witness operator (support, 3 qubits)
[5, 23, 29]
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 · 2026-11-01
witness operator (support, 3 qubits)
[2, 31, 32]
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 8–27 (mean 14.333) · H_Z 8–27 (mean 14.333)
qubit degrees H_X 1–4 (mean 2.389) · H_Z 1–4 (mean 2.389)
trapping sets H_X (1,1)×9 (2,1)×114 (3,0)×148 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 9 (1,2): 21 (1,3): 18 (1,4): 6 (2,1): 114 (2,2): 270 (2,3): 348 (2,4): 180 (3,0): 148 (3,1): 1338 (3,2): 3294 (3,3): 4227 (3,4): 3576 (3,5): 2130 (3,6): 740
trapping sets H_Z (1,1)×9 (2,1)×114 (3,0)×148 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 9 (1,2): 21 (1,3): 18 (1,4): 6 (2,1): 114 (2,2): 270 (2,3): 348 (2,4): 180 (3,0): 148 (3,1): 1338 (3,2): 3294 (3,3): 4227 (3,4): 3576 (3,5): 2130 (3,6): 740
witness diameter X 3.1623 · Z 3.1623 (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/54-36-3/ · 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)
[4487, 4493, 4866]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[8343, 8346, 8389]
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/56-36-4.json: the parent's coordinates with the entries of the removed qubits (parent indices 0, 7) dropped, since this peel chain keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 6.325
X checkZ checkqubit site (28)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 32 nearest-neighbor SWAPs per round in total, at most 5 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-PEEL transform of codes/56-36-4.json
model Beam 1.0 (claimed, not verified)
date 2026-11-01
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

[[54,36,3]] — priced calculus E-PEEL transform of 56-36-4.json

Direction & hypothesis

Weight-27 unrestricted CSS cell. Transform a connected CSS parent (56-36-4.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 and depth-2 deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev. Parent: codes/56-36-4.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/56-36-4.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/56-36-4.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 9 (max weight 27) · Z-checks 9 (max weight 27)
H_X (9 checks, sparse supports)
[6, 7, 8, 9, 10, 11, 12, 13] [14, 15, 16, 17, 18, 19, 20, 21] [22, 23, 24, 25, 26, 27, 28, 29] [30, 31, 32, 33, 34, 35, 36, 37] [38, 39, 40, 41, 42, 43, 44, 45] [46, 47, 48, 49, 50, 51, 52, 53] [1, 2, 5, 6, 8, 9, 12, 14, 16, 17, 20, 22, 24, 25, 28, 30, 32, 33, 36, 38, 40, 41, 44, 46, 48, 49, 52] [0, 2, 4, 6, 7, 9, 11, 14, 15, 17, 19, 22, 23, 25, 27, 30, 31, 33, 35, 38, 39, 41, 43, 46, 47, 49, 51] [0, 1, 3, 6, 7, 8, 10, 14, 15, 16, 18, 22, 23, 24, 26, 30, 31, 32, 34, 38, 39, 40, 42, 46, 47, 48, 50]
H_Z (9 checks, sparse supports)
[6, 7, 8, 9, 10, 11, 12, 13] [14, 15, 16, 17, 18, 19, 20, 21] [22, 23, 24, 25, 26, 27, 28, 29] [30, 31, 32, 33, 34, 35, 36, 37] [38, 39, 40, 41, 42, 43, 44, 45] [46, 47, 48, 49, 50, 51, 52, 53] [0, 3, 4, 7, 10, 11, 13, 15, 18, 19, 21, 23, 26, 27, 29, 31, 34, 35, 37, 39, 42, 43, 45, 47, 50, 51, 53] [1, 3, 5, 8, 10, 12, 13, 16, 18, 20, 21, 24, 26, 28, 29, 32, 34, 36, 37, 40, 42, 44, 45, 48, 50, 52, 53] [2, 4, 5, 9, 11, 12, 13, 17, 19, 20, 21, 25, 27, 28, 29, 33, 35, 36, 37, 41, 43, 44, 45, 49, 51, 52, 53]
Code ID 54-36-3 · download JSON · raw on GitHub