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[[59,23,3]] d =
n
59
k
23
d
3
kd²/n
3.50847
w
6
X/Z
1
g
0.0121
r
4.1231
layers
2
swaps
112

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 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)
[1, 19, 47]
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)
[10, 13, 31]
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 6 · H_Z 6
qubit degrees H_X 1–2 (mean 1.831) · H_Z 1–2 (mean 1.831)
trapping sets H_X (1,1)×10 (2,1)×50 (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): 10 (1,2): 49 (2,1): 50 (2,2): 220 (3,0): 9 (3,1): 232 (3,2): 984 (3,3): 100 (3,4): 260
trapping sets H_Z (1,1)×10 (2,1)×50 (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): 10 (1,2): 49 (2,1): 50 (2,2): 220 (3,0): 9 (3,1): 232 (3,2): 984 (3,3): 100 (3,4): 260
witness diameter X 4.1231 · Z 1.0 (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/60-22-4.json: the parent's coordinates with the entry of the removed qubit (parent index 28) 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 = 4.123
X checkZ checkqubit site (30)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 112 nearest-neighbor SWAPs per round in total, at most 6 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/60-22-4.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 ≤ 6 (computed)

How this code was found

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

[[59,23,3]] — priced calculus E-PUNCTURE transform of 60-22-4.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (60-22-4.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/60-22-4.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/60-22-4.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/60-22-4.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 18 (max weight 6) · Z-checks 18 (max weight 6)
H_X (18 checks, sparse supports)
[3, 36, 44, 45, 49, 58] [1, 8, 11, 12, 15, 50] [16, 17, 23, 28, 35, 39] [20, 22, 28, 33, 36, 54] [10, 13, 20, 34, 42, 55] [3, 5, 6, 17, 52, 57] [18, 27, 30, 41, 46, 57] [7, 12, 14, 23, 47, 49] [5, 24, 29, 40, 51, 53] [4, 19, 35, 44, 55, 56] [24, 26, 27, 37, 39, 45] [1, 4, 25, 30, 40, 54] [8, 13, 31, 37, 38, 52] [9, 16, 21, 32, 50, 58] [2, 6, 11, 26, 42, 43] [0, 15, 18, 19, 29, 33] [9, 34, 41, 47, 48, 53] [0, 7, 21, 25, 38, 43]
H_Z (18 checks, sparse supports)
[3, 7, 25, 30, 49, 57] [1, 12, 20, 34, 47, 54] [8, 9, 41, 50, 52, 57] [13, 20, 21, 36, 38, 58] [7, 14, 24, 26, 43, 51] [4, 5, 6, 40, 42, 55] [18, 22, 27, 31, 33, 37] [16, 28, 30, 32, 46, 54] [19, 29, 44, 47, 49, 53] [2, 9, 26, 45, 48, 58] [5, 17, 22, 28, 48, 53] [3, 6, 11, 15, 33, 36] [2, 11, 12, 23, 35, 56] [15, 16, 24, 29, 39, 50] [10, 27, 44, 45, 46, 55] [4, 25, 35, 37, 38, 39] [0, 18, 34, 41, 42, 43] [10, 13, 14, 17, 23, 52]
Code ID 59-23-3 · download JSON · raw on GitHub