← back to the board
[[125,12,9]] d ≤
n
125
k
12
d
9
kd²/n
7.776
w
6
X/Z
1.11
g
0.0057
r
6.0828
layers
2
swaps
556

Share this result

Distance

X/Z asymmetry 1.11 · d_X ≤ 10, d_Z ≤ 9 · 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 10 · witness weight 10 (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, 10 qubits)
[17, 40, 52, 61, 62, 63, 79, 119, 120, 124]
d_Z 9 · witness weight 9 (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, 9 qubits)
[22, 51, 57, 62, 63, 76, 89, 98, 124]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 5–6 (mean 5.965)
qubit degrees H_X 1–3 (mean 2.688) · H_Z 1–3 (mean 2.72)
trapping sets H_X (1,1)×6 (2,1)×2 (3,1)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 6 (1,2): 27 (1,3): 92 (2,1): 2 (2,2): 34 (2,3): 256 (2,4): 548 (3,1): 5 (3,2): 114 (3,3): 639 (3,4): 2491 (3,5): 4054 (3,6): 484 (3,7): 552
trapping sets H_Z (1,1)×6 (2,1)×2 (3,1)×5 (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): 23 (1,3): 96 (2,1): 2 (2,2): 32 (2,3): 220 (2,4): 591 (3,1): 5 (3,2): 93 (3,3): 574 (3,4): 2213 (3,5): 4500 (3,6): 420 (3,7): 624
witness diameter X 8.0 · Z 6.3246 (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/126-12-10.json: the parent's coordinates with the entry of the removed qubit (parent index 57) 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 = 6.083
X checkZ checkqubit site (63)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 556 nearest-neighbor SWAPs per round in total, at most 11 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/126-12-10.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 ≤ 6 (computed)

How this code was found

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

[[125,12,9]] — priced calculus E-PEEL transform of codes/126-12-10.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/126-12-10.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/126-12-10.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/126-12-10.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/126-12-10.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 56 (max weight 6) · Z-checks 57 (max weight 6)
H_X (56 checks, sparse supports)
[0, 1, 57, 62, 75, 103] [1, 2, 58, 63, 76, 104] [2, 3, 59, 64, 77, 105] [3, 4, 60, 65, 78, 106] [4, 5, 61, 66, 79, 107] [0, 5, 6, 67, 80, 108] [1, 6, 7, 68, 81, 109] [2, 7, 8, 69, 82, 110] [3, 8, 9, 70, 83, 111] [4, 9, 10, 71, 84, 112] [5, 10, 11, 72, 85, 113] [6, 11, 12, 73, 86, 114] [7, 12, 13, 74, 87, 115] [8, 13, 14, 75, 88, 116] [9, 14, 15, 76, 89, 117] [10, 15, 16, 77, 90, 118] [11, 16, 17, 78, 91, 119] [12, 17, 18, 79, 92, 120] [13, 18, 19, 80, 93, 121] [14, 19, 20, 81, 94, 122] [15, 20, 21, 82, 95, 123] [16, 21, 22, 83, 96, 124] [17, 22, 23, 62, 84, 97] [18, 23, 24, 63, 85, 98] [19, 24, 25, 64, 86, 99] [20, 25, 26, 65, 87, 100] [21, 26, 27, 66, 88, 101] [22, 27, 28, 67, 89, 102] [23, 28, 29, 68, 90, 103] [24, 29, 30, 69, 91, 104] [25, 30, 31, 70, 92, 105] [26, 31, 32, 71, 93, 106] [27, 32, 33, 72, 94, 107] [28, 33, 34, 73, 95, 108] [29, 34, 35, 74, 96, 109] [30, 35, 36, 75, 97, 110] [31, 36, 37, 76, 98, 111] [32, 37, 38, 77, 99, 112] [33, 38, 39, 78, 100, 113] [34, 39, 40, 79, 101, 114] [35, 40, 41, 80, 102, 115] [36, 41, 42, 81, 103, 116] [37, 42, 43, 82, 104, 117] [38, 43, 44, 83, 105, 118] [39, 44, 45, 84, 106, 119] [40, 45, 46, 85, 107, 120] [41, 46, 47, 86, 108, 121] [42, 47, 48, 87, 109, 122] [43, 48, 49, 88, 110, 123] [44, 49, 50, 89, 111, 124] [45, 50, 51, 62, 90, 112] [46, 51, 52, 63, 91, 113] [47, 52, 53, 64, 92, 114] [48, 53, 54, 65, 93, 115] [49, 54, 55, 66, 94, 116] [50, 55, 56, 67, 95, 117]
H_Z (57 checks, sparse supports)
[0, 22, 50, 62, 67, 124] [1, 23, 51, 62, 63, 68] [2, 24, 52, 63, 64, 69] [3, 25, 53, 64, 65, 70] [4, 26, 54, 65, 66, 71] [5, 27, 55, 66, 67, 72] [6, 28, 56, 67, 68, 73] [7, 29, 68, 69, 74] [8, 30, 57, 69, 70, 75] [9, 31, 58, 70, 71, 76] [10, 32, 59, 71, 72, 77] [11, 33, 60, 72, 73, 78] [12, 34, 61, 73, 74, 79] [0, 13, 35, 74, 75, 80] [1, 14, 36, 75, 76, 81] [2, 15, 37, 76, 77, 82] [3, 16, 38, 77, 78, 83] [4, 17, 39, 78, 79, 84] [5, 18, 40, 79, 80, 85] [6, 19, 41, 80, 81, 86] [7, 20, 42, 81, 82, 87] [8, 21, 43, 82, 83, 88] [9, 22, 44, 83, 84, 89] [10, 23, 45, 84, 85, 90] [11, 24, 46, 85, 86, 91] [12, 25, 47, 86, 87, 92] [13, 26, 48, 87, 88, 93] [14, 27, 49, 88, 89, 94] [15, 28, 50, 89, 90, 95] [16, 29, 51, 90, 91, 96] [17, 30, 52, 91, 92, 97] [18, 31, 53, 92, 93, 98] [19, 32, 54, 93, 94, 99] [20, 33, 55, 94, 95, 100] [21, 34, 56, 95, 96, 101] [22, 35, 96, 97, 102] [23, 36, 57, 97, 98, 103] [24, 37, 58, 98, 99, 104] [25, 38, 59, 99, 100, 105] [26, 39, 60, 100, 101, 106] [27, 40, 61, 101, 102, 107] [0, 28, 41, 102, 103, 108] [1, 29, 42, 103, 104, 109] [2, 30, 43, 104, 105, 110] [3, 31, 44, 105, 106, 111] [4, 32, 45, 106, 107, 112] [5, 33, 46, 107, 108, 113] [6, 34, 47, 108, 109, 114] [7, 35, 48, 109, 110, 115] [8, 36, 49, 110, 111, 116] [9, 37, 50, 111, 112, 117] [10, 38, 51, 112, 113, 118] [11, 39, 52, 113, 114, 119] [12, 40, 53, 114, 115, 120] [13, 41, 54, 115, 116, 121] [14, 42, 55, 116, 117, 122] [15, 43, 56, 117, 118, 123]
Code ID 125-12-9 · download JSON · raw on GitHub