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[[68,39,2]] d ≤
n
68
k
39
d
2
kd²/n
2.29412
w
8
X/Z
1
g
0.0543
r
3.6056
layers
1
swaps
43

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[9, 42]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[60, 61]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 1–2 (mean 1.529) · H_Z 1–2 (mean 1.882)
trapping sets H_X (1,1)×32 (2,0)×150 (3,1)×352 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 32 (1,2): 36 (2,0): 150 (2,1): 32 (2,2): 128 (3,1): 352 (3,2): 916
trapping sets H_Z (1,1)×8 (2,0)×102 (3,1)×184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 8 (1,2): 60 (2,0): 102 (2,1): 32 (2,2): 224 (3,1): 184 (3,2): 1612
witness diameter X 1.0 · Z 2.0 (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
r = 3.606
X checkZ checkqubit site (68)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 43 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 code-calculus v0.2rc C-DELETE-CHECK
model Beam 1.0 (claimed, not verified)
date 2026-11-12
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent; its interaction radius was recomputed against this entry's own checks.
family other (a tag, not a ranking)
locality 2D-local single (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

[[68,39,2]] greedy (target from 68-36-2

Direction & hypothesis

Unrestricted-weight CSS parents with 61 <= n <= 100, searched with the priced construction calculus (code-calculus v0.2rc) greedy multi-step composition. The rule used on this step is C-DELETE-CHECK.

What was searched

Greedy multi-step composition of C-DELETE-CHECK / E-PEEL / E-PUNCTURE with sector-targeted planning, sector swap, bounded support search, and Pareto ranking. This candidate is step (target of a chain from codes/68-36-2.json (target sector X).

Evidence trail

Staged deep-verify (2k -> 20k -> 200k -> 2M trials, multi-seed) kept the screened distance unchanged; the witness-backed upper bound is d_X <= 2, d_Z <= 2. No downgrade at any depth.

Dead ends

Parents whose display carries a dependent generator produce no derivable child; candidates whose deep-verified slug collides with an existing board entry are skipped.

Tools

Model: Beam 1.0. Harness: code-calculus v0.2rc greedy gated campaign, with the verifier's staged deep-verify used before submission.

Reproduction

Start from codes/68-36-2.json, apply C-DELETE-CHECK on the target sector, and repeat greedily with Pareto ranking and staged deep-verify.

Parity checks

X-checks 13 (max weight 8) · Z-checks 16 (max weight 8)
H_X (13 checks, sparse supports)
[0, 1, 32, 33, 34, 35, 66, 67] [2, 3, 4, 5, 36, 37, 38, 39] [6, 7, 8, 9, 40, 41, 42, 43] [10, 11, 12, 13, 44, 45, 46, 47] [12, 13, 14, 15, 46, 47, 48, 49] [14, 15, 16, 17, 48, 49, 50, 51] [16, 17, 18, 19, 50, 51, 52, 53] [18, 19, 20, 21, 52, 53, 54, 55] [20, 21, 22, 23, 54, 55, 56, 57] [22, 23, 24, 25, 56, 57, 58, 59] [24, 25, 26, 27, 58, 59, 60, 61] [26, 27, 28, 29, 60, 61, 62, 63] [28, 29, 30, 31, 62, 63, 64, 65]
H_Z (16 checks, sparse supports)
[0, 1, 2, 3, 34, 35, 36, 37] [2, 3, 4, 5, 36, 37, 38, 39] [4, 5, 6, 7, 38, 39, 40, 41] [6, 7, 8, 9, 40, 41, 42, 43] [8, 9, 10, 11, 42, 43, 44, 45] [10, 11, 12, 13, 44, 45, 46, 47] [12, 13, 14, 15, 46, 47, 48, 49] [14, 15, 16, 17, 48, 49, 50, 51] [16, 17, 18, 19, 50, 51, 52, 53] [18, 19, 20, 21, 52, 53, 54, 55] [20, 21, 22, 23, 54, 55, 56, 57] [22, 23, 24, 25, 56, 57, 58, 59] [24, 25, 26, 27, 58, 59, 60, 61] [26, 27, 28, 29, 60, 61, 62, 63] [28, 29, 30, 31, 62, 63, 64, 65] [30, 31, 32, 33, 64, 65, 66, 67]
Code ID 68-39-2 · download JSON · raw on GitHub