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[[111,12,11]] d ≤
n
111
k
12
d
11
kd²/n
13.0811
w
8
X/Z
1.09
g
0.0065
r
6.7082
layers
2
swaps
634

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Distance

X/Z asymmetry 1.09 · d_X ≤ 12, d_Z ≤ 11 · 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 12 · witness weight 12 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 12 qubits)
[3, 18, 19, 32, 48, 49, 59, 64, 82, 100, 103, 110]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 11 qubits)
[1, 6, 16, 23, 24, 34, 47, 61, 92, 97, 104]
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 7–8 (mean 7.92)
qubit degrees H_X 1–5 (mean 3.532) · H_Z 1–5 (mean 3.568)
trapping sets H_X (1,1)×2 (2,1)×1 (3,2)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 15 (1,3): 44 (1,4): 22 (1,5): 28 (2,1): 1 (2,2): 8 (2,3): 85 (2,4): 205 (2,5): 301 (2,6): 379 (2,7): 196 (2,8): 134 (3,2): 12 (3,3): 116 (3,4): 701 (3,5): 1846 (3,6): 3727 (3,7): 5189 (3,8): 5139 (3,9): 4299 (3,10): 2007 (3,11): 1065 (3,12): 132 (3,13): 51
trapping sets H_Z (1,1)×3 (2,1)×1 (3,2)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 3 (1,2): 12 (1,3): 45 (1,4): 21 (1,5): 30 (2,1): 1 (2,2): 10 (2,3): 64 (2,4): 202 (2,5): 286 (2,6): 396 (2,7): 201 (2,8): 149 (3,2): 14 (3,3): 95 (3,4): 553 (3,5): 1690 (3,6): 3451 (3,7): 5251 (3,8): 5125 (3,9): 4613 (3,10): 2178 (3,11): 1197 (3,12): 148 (3,13): 57
witness diameter X 7.2801 · Z 7.8102 (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/112-12-12.json: the parent's coordinates with the entry of the removed qubit (parent index 45) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 6.708
X checkZ checkqubit site (56)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 634 nearest-neighbor SWAPs per round in total, at most 12 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/112-12-12.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
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

[[111,12,11]] — priced calculus E-PEEL transform of codes/112-12-12.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (codes/112-12-12.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

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/112-12-12.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, 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. Parent entry: codes/112-12-12.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/112-12-12.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 49 (max weight 8) · Z-checks 50 (max weight 8)
H_X (49 checks, sparse supports)
[0, 32, 45, 55, 56, 82, 95, 105] [2, 26, 50, 57, 68, 88, 89, 109] [1, 3, 40, 58, 59, 61, 90, 103] [4, 41, 52, 59, 62, 73, 97, 104] [5, 39, 47, 60, 64, 92, 93, 110] [0, 6, 51, 58, 61, 62, 94, 106] [7, 36, 49, 59, 62, 77, 101, 107] [5, 8, 34, 63, 64, 76, 96, 97] [9, 35, 54, 60, 64, 79, 98, 103] [6, 10, 47, 57, 65, 66, 69, 98] [7, 11, 48, 55, 66, 70, 81, 109] [2, 12, 46, 67, 68, 72, 100, 101] [13, 30, 53, 57, 68, 83, 102, 106] [3, 14, 54, 60, 65, 69, 70, 102] [4, 15, 44, 56, 66, 70, 85, 110] [12, 16, 42, 55, 71, 72, 84, 104] [13, 17, 43, 57, 67, 72, 87, 105] [2, 14, 18, 63, 73, 74, 77, 105] [15, 19, 53, 58, 64, 74, 78, 89] [8, 20, 52, 56, 75, 76, 80, 107] [9, 21, 38, 60, 63, 76, 91, 108] [10, 13, 22, 67, 73, 77, 78, 108] [11, 23, 50, 61, 68, 74, 78, 93] [0, 20, 24, 58, 79, 80, 92, 109] [21, 25, 49, 59, 63, 75, 80, 95] [8, 22, 26, 59, 71, 81, 82, 85] [9, 23, 27, 65, 72, 82, 86, 97] [7, 16, 28, 61, 83, 84, 88, 110] [17, 29, 45, 62, 67, 71, 84, 99] [18, 21, 30, 62, 75, 81, 85, 86] [5, 19, 31, 69, 76, 82, 86, 101] [3, 28, 32, 64, 65, 87, 88, 100] [4, 29, 33, 66, 71, 83, 88, 103] [16, 30, 34, 66, 79, 89, 90, 93] [17, 31, 35, 55, 73, 80, 90, 94] [15, 24, 36, 68, 69, 91, 92, 96] [1, 25, 37, 70, 75, 79, 92, 106] [26, 29, 38, 70, 83, 89, 93, 94] [12, 27, 39, 56, 77, 84, 90, 94] [10, 36, 40, 72, 73, 95, 96, 107] [11, 37, 41, 57, 74, 79, 91, 96] [24, 38, 42, 74, 87, 97, 98, 101] [25, 39, 43, 58, 81, 88, 98, 102] [23, 32, 44, 76, 77, 99, 100, 104] [34, 37, 45, 78, 91, 97, 101, 102] [20, 35, 46, 61, 85, 92, 98, 102] [18, 44, 47, 80, 81, 103, 104, 110] [33, 46, 49, 65, 89, 96, 105, 108] [31, 40, 50, 84, 85, 106, 107, 109]
H_Z (50 checks, sparse supports)
[0, 1, 11, 16, 35, 55, 61, 79] [0, 1, 15, 20, 39, 56, 58, 92] [2, 10, 13, 17, 41, 57, 67, 73] [3, 6, 19, 24, 43, 58, 69, 87] [3, 4, 7, 25, 26, 59, 70, 88] [5, 9, 14, 21, 60, 63, 86] [3, 6, 23, 28, 46, 61, 65, 100] [4, 6, 7, 29, 30, 62, 66, 83] [8, 18, 21, 25, 48, 63, 75, 81] [5, 8, 9, 19, 32, 64, 76, 82] [10, 14, 27, 32, 49, 65, 77, 95] [10, 11, 15, 33, 34, 66, 78, 96] [12, 17, 22, 29, 51, 67, 71, 94] [2, 12, 13, 23, 36, 68, 72, 77] [10, 14, 31, 36, 52, 69, 73, 107] [11, 14, 15, 37, 38, 70, 74, 91] [16, 26, 29, 33, 53, 71, 83, 89] [12, 16, 17, 27, 40, 72, 84, 90] [4, 18, 22, 35, 40, 73, 85, 103] [18, 19, 23, 41, 42, 74, 86, 104] [20, 25, 30, 37, 54, 75, 79, 102] [8, 20, 21, 31, 44, 76, 80, 85] [7, 18, 22, 39, 44, 77, 81, 110] [19, 22, 23, 45, 78, 82, 99] [9, 24, 34, 37, 41, 79, 91, 97] [20, 24, 25, 35, 47, 80, 92, 98] [11, 26, 30, 43, 47, 57, 81, 93] [0, 26, 27, 31, 48, 82, 94, 109] [13, 28, 33, 38, 83, 87, 108] [16, 28, 29, 39, 50, 84, 88, 93] [15, 26, 30, 46, 50, 68, 85, 89] [1, 27, 30, 31, 51, 86, 90, 106] [17, 32, 42, 48, 55, 87, 99] [2, 28, 32, 33, 43, 88, 100, 105] [2, 19, 34, 38, 49, 63, 89, 101] [3, 34, 35, 39, 53, 64, 90, 102] [21, 36, 41, 45, 51, 62, 91, 95] [5, 24, 36, 37, 46, 92, 96, 101] [5, 23, 34, 38, 52, 76, 93, 97] [6, 35, 38, 39, 54, 60, 94, 98] [0, 25, 40, 51, 53, 58, 95, 106] [8, 36, 40, 41, 49, 59, 96, 107] [4, 8, 27, 42, 45, 56, 71, 97] [9, 10, 42, 43, 46, 72, 98, 108] [1, 29, 44, 48, 54, 70, 99, 103] [7, 12, 31, 42, 45, 55, 84, 101] [13, 14, 43, 45, 46, 67, 102, 105] [3, 9, 33, 47, 54, 60, 65, 103] [0, 17, 18, 49, 52, 62, 80, 105] [6, 13, 37, 50, 53, 57, 78, 106]
Code ID 111-12-11 · download JSON · raw on GitHub