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[[29,5,4]] d =
n
29
k
5
d
4
kd²/n
2.75862
w
5
X/Z
1
g
0.0108
r
4.0
layers
2
swaps
49

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (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, 4 qubits)
[4, 15, 18, 19]
d_Z 4 · witness weight 4 (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, 4 qubits)
[5, 7, 19, 22]
certificate exact, d = 4 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI
X: no logical < 4 exists; Z: no logical < 4 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 1–3 (mean 2.069) · H_Z 1–3 (mean 2.069)
trapping sets H_X (1,1)×5 (2,1)×9 (3,1)×40 (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): 7 (2,1): 9 (2,2): 49 (2,3): 51 (2,4): 11 (3,1): 40 (3,2): 153 (3,3): 250 (3,4): 150 (3,5): 63 (3,6): 23 (3,7): 1
trapping sets H_Z (1,1)×6 (2,1)×10 (3,1)×42 (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): 15 (1,3): 8 (2,1): 10 (2,2): 44 (2,3): 50 (2,4): 16 (3,1): 42 (3,2): 133 (3,3): 249 (3,4): 160 (3,5): 75 (3,6): 25 (3,7): 4
witness diameter X 3.1623 · 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)

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/30-4-5.json: the parent's coordinates with the entry of the removed qubit (parent index 14) 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
X checkZ checkqubit site (15)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 49 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-PUNCTURE transform of codes/30-4-5.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 ≤ 6 (computed)

How this code was found

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

[[29,5,4]] — priced-calculus transform of 30-4-5.json

Direction & hypothesis

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