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[[123,10,9]] d ≤
n
123
k
10
d
9
kd²/n
6.58537
w
6
X/Z
1.11
g
0.0078
r
5.3852
layers
2
swaps
495

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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)
[24, 32, 40, 44, 48, 56, 59, 62, 119, 121]
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)
[42, 45, 47, 48, 51, 52, 53, 55, 100]
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 2.732) · H_Z 1–3 (mean 2.764)
trapping sets H_X (1,1)×4 (2,1)×6 (3,1)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 4 (1,2): 25 (1,3): 94 (2,1): 6 (2,2): 31 (2,3): 210 (2,4): 593 (3,1): 8 (3,2): 98 (3,3): 591 (3,4): 2058 (3,5): 4638 (3,6): 353 (3,7): 667
trapping sets H_Z (1,1)×5 (2,1)×4 (3,1)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 5 (1,2): 19 (1,3): 99 (2,1): 4 (2,2): 28 (2,3): 172 (2,4): 641 (3,1): 7 (3,2): 78 (3,3): 521 (3,4): 1756 (3,5): 5128 (3,6): 308 (3,7): 734
witness diameter X 7.2111 · Z 8.0623 (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/124-10-10.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 = 5.385
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 495 nearest-neighbor SWAPs per round in total, at most 9 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/124-10-10.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

[[123,10,9]] — priced calculus E-PEEL transform of codes/124-10-10.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/124-10-10.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/124-10-10.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/124-10-10.json.

Tools

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

Reproduction

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