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[[53,9,5]] d =
n
53
k
9
d
5
kd²/n
4.24528
w
6
X/Z
1
g
0.0166
r
5.6569
layers
1
swaps
218

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 5 · witness weight 5 (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, 5 qubits)
[7, 9, 15, 44, 51]
d_Z 5 · witness weight 5 (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, 5 qubits)
[10, 18, 24, 45, 46]
certificate exact, d = 5 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI
X: no logical < 5 exists; Z: no logical < 5 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 6 · H_Z 6
qubit degrees H_X 1–3 (mean 2.491) · H_Z 1–3 (mean 2.491)
trapping sets H_X (1,1)×5 (2,1)×12 (3,1)×34 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 5 (1,2): 17 (1,3): 31 (2,1): 12 (2,2): 50 (2,3): 116 (2,4): 133 (3,1): 34 (3,2): 132 (3,3): 490 (3,4): 937 (3,5): 843 (3,6): 118 (3,7): 86
trapping sets H_Z (1,1)×4 (2,1)×9 (3,1)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 4 (1,2): 19 (1,3): 30 (2,1): 9 (2,2): 54 (2,3): 127 (2,4): 121 (3,1): 25 (3,2): 160 (3,3): 515 (3,4): 964 (3,5): 758 (3,6): 136 (3,7): 65
witness diameter X 4.0 · Z 6.4031 (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/54-8-6.json: the parent's coordinates with the entry of the removed qubit (parent index 46) 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 = 5.657
X checkZ checkqubit site (53)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 218 nearest-neighbor SWAPs per round in total, at most 9 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/54-8-6.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 ≤ 6 (computed)

How this code was found

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

[[53,9,5]] — priced calculus E-PUNCTURE transform of 54-8-6.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (54-8-6.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/54-8-6.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/54-8-6.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/54-8-6.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 22 (max weight 6) · Z-checks 22 (max weight 6)
H_X (22 checks, sparse supports)
[0, 3, 5, 6, 12, 40] [1, 3, 4, 7, 13, 41] [2, 4, 5, 8, 14, 39] [7, 9, 10, 13, 19, 46] [8, 10, 11, 14, 20, 45] [12, 15, 17, 18, 24, 51] [13, 15, 16, 19, 25, 52] [14, 16, 17, 20, 26, 50] [4, 18, 21, 23, 24, 30] [5, 19, 21, 22, 25, 31] [3, 20, 22, 23, 26, 32] [10, 24, 27, 29, 30, 36] [11, 25, 27, 28, 31, 37] [9, 26, 28, 29, 32, 38] [16, 30, 33, 35, 36, 42] [17, 31, 33, 34, 37, 43] [15, 32, 34, 35, 38, 44] [22, 36, 39, 41, 42, 47] [23, 37, 39, 40, 43, 48] [21, 38, 40, 41, 44, 49] [0, 28, 42, 45, 46, 47] [0, 6, 34, 47, 50, 52]
H_Z (22 checks, sparse supports)
[0, 1, 3, 20, 45, 50] [0, 2, 5, 19, 46, 52] [3, 6, 7, 9, 26, 50] [4, 7, 8, 10, 24, 51] [5, 6, 8, 11, 25, 52] [3, 9, 12, 13, 15, 32] [4, 10, 13, 14, 16, 30] [5, 11, 12, 14, 17, 31] [9, 15, 18, 19, 21, 38] [10, 16, 19, 20, 22, 36] [11, 17, 18, 20, 23, 37] [15, 21, 24, 25, 27, 44] [16, 22, 25, 26, 28, 42] [21, 27, 30, 31, 33, 49] [22, 28, 31, 32, 34, 47] [23, 29, 30, 32, 35, 48] [2, 27, 33, 36, 37, 39] [0, 28, 34, 37, 38, 40] [1, 29, 35, 36, 38, 41] [8, 33, 39, 42, 43, 45] [7, 35, 41, 42, 44, 46] [14, 39, 45, 47, 48, 50]
Code ID 53-9-5 · download JSON · raw on GitHub