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[[159,40,3]] d ≤
n
159
k
40
d
3
kd²/n
2.26415
w
6
X/Z
1.33
g
0.0078
r
4.1231
layers
2
swaps
165

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Distance

X/Z asymmetry 1.33 · d_X ≤ 4, d_Z ≤ 3 · 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 4 · witness weight 4 (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, 4 qubits)
[29, 46, 121, 138]
d_Z 3 · witness weight 3 (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, 3 qubits)
[25, 30, 97]
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 6 · H_Z 5–6 (mean 5.95)
qubit degrees H_X 1–4 (mean 2.226) · H_Z 1–4 (mean 2.245)
trapping sets H_X (1,1)×41 (2,1)×85 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 41 (1,2): 59 (1,3): 41 (1,4): 18 (2,1): 85 (2,2): 182 (2,3): 259 (2,4): 193 (2,5): 91 (2,6): 17 (3,0): 9 (3,1): 360 (3,2): 848 (3,3): 1216 (3,4): 1404 (3,5): 1094 (3,6): 521 (3,7): 247 (3,8): 68
trapping sets H_Z (1,1)×40 (2,1)×80 (3,1)×354 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 40 (1,2): 60 (1,3): 39 (1,4): 20 (2,1): 80 (2,2): 177 (2,3): 255 (2,4): 196 (2,5): 98 (2,6): 20 (3,1): 354 (3,2): 803 (3,3): 1192 (3,4): 1399 (3,5): 1128 (3,6): 542 (3,7): 272 (3,8): 79
witness diameter X 2.0 · Z 2.2361 (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/160-40-4.json: the parent's coordinates with the entry of the removed qubit (parent index 1) 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 = 4.123
X checkZ checkqubit site (80)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 165 nearest-neighbor SWAPs per round in total, at most 4 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/160-40-4.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

[[159,40,3]] — priced calculus E-PEEL transform of codes/160-40-4.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/160-40-4.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/160-40-4.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/160-40-4.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/160-40-4.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 59 (max weight 6) · Z-checks 60 (max weight 6)
H_X (59 checks, sparse supports)
[0, 10, 79, 90, 147, 156] [0, 1, 79, 81, 154, 158] [2, 24, 82, 88, 102, 104] [1, 3, 81, 83, 147, 156] [4, 26, 84, 106, 131, 146] [5, 7, 80, 85, 135, 149] [6, 16, 79, 83, 86, 112] [7, 17, 85, 87, 123, 139] [8, 18, 88, 114, 140, 152] [2, 9, 82, 89, 98, 114] [4, 11, 84, 91, 142, 153] [12, 40, 92, 101, 118, 120] [9, 13, 88, 89, 93, 102] [14, 42, 80, 87, 94, 122] [11, 15, 91, 95, 131, 146] [16, 20, 81, 86, 90, 96] [18, 22, 88, 98, 150, 157] [19, 31, 82, 93, 99, 128] [20, 32, 79, 83, 96, 100] [21, 33, 84, 95, 101, 130] [22, 34, 98, 102, 140, 152] [12, 23, 92, 103, 113, 130] [14, 25, 85, 94, 97, 105] [27, 56, 107, 117, 134, 136] [23, 28, 101, 103, 108, 118] [29, 58, 86, 100, 109, 138] [25, 30, 80, 87, 105, 110] [31, 36, 89, 99, 104, 111] [33, 38, 91, 101, 106, 113] [35, 47, 92, 108, 115, 143] [36, 48, 82, 93, 111, 116] [37, 49, 94, 110, 117, 145] [38, 50, 84, 95, 113, 118] [27, 39, 107, 119, 129, 145] [29, 41, 96, 109, 112, 121] [43, 69, 123, 133, 148, 149] [39, 44, 117, 119, 124, 134] [45, 71, 99, 116, 125, 151] [41, 46, 86, 100, 121, 126] [47, 52, 103, 115, 120, 127] [49, 54, 105, 117, 122, 129] [51, 62, 107, 124, 131, 153] [52, 63, 92, 108, 127, 132] [53, 64, 109, 126, 133, 155] [54, 65, 94, 110, 129, 134] [43, 55, 123, 135, 144, 155] [45, 57, 111, 125, 128, 137] [55, 59, 133, 135, 139, 148] [60, 77, 115, 132, 140, 157] [57, 61, 99, 116, 137, 141] [62, 66, 119, 131, 136, 142] [64, 68, 121, 133, 138, 144] [66, 73, 107, 124, 142, 146] [67, 74, 125, 141, 147, 158] [68, 75, 109, 126, 144, 148] [60, 70, 127, 140, 143, 150] [70, 72, 115, 132, 150, 152] [74, 76, 137, 147, 151, 154] [76, 78, 125, 141, 154, 156]
H_Z (60 checks, sparse supports)
[0, 1, 6, 20, 79, 81] [5, 14, 30, 80, 97] [1, 3, 16, 32, 81, 83] [2, 9, 19, 36, 82, 89] [3, 6, 10, 20, 83, 90] [4, 11, 21, 38, 84, 91] [5, 7, 25, 42, 80, 85] [6, 16, 29, 46, 86, 112] [7, 14, 17, 30, 85, 87] [2, 8, 13, 18, 88, 114] [9, 13, 31, 48, 89, 93] [11, 15, 33, 50, 91, 95] [12, 23, 35, 52, 92, 103] [13, 19, 24, 36, 93, 104] [14, 25, 37, 54, 94, 105] [15, 21, 26, 38, 95, 106] [16, 20, 41, 58, 86, 96] [9, 18, 22, 24, 88, 98] [19, 31, 45, 61, 99, 128] [20, 29, 32, 46, 96, 100] [12, 21, 28, 33, 101, 130] [2, 13, 22, 34, 98, 102] [23, 28, 47, 63, 103, 108] [25, 30, 49, 65, 105, 110] [27, 39, 51, 66, 107, 119] [28, 35, 40, 52, 108, 120] [29, 41, 53, 68, 109, 121] [30, 37, 42, 54, 110, 122] [31, 36, 57, 71, 99, 111] [23, 33, 38, 40, 101, 113] [35, 47, 60, 72, 115, 143] [36, 45, 48, 61, 111, 116] [27, 37, 44, 49, 117, 145] [12, 28, 38, 50, 113, 118] [39, 44, 62, 73, 119, 124] [41, 46, 64, 75, 121, 126] [7, 43, 55, 123, 135] [44, 51, 56, 66, 124, 136] [45, 57, 67, 76, 125, 137] [46, 53, 58, 68, 126, 138] [47, 52, 70, 77, 115, 127] [39, 49, 54, 56, 117, 129] [4, 15, 51, 62, 131, 153] [52, 60, 63, 72, 127, 132] [43, 53, 59, 64, 133, 155] [27, 44, 54, 65, 129, 134] [5, 17, 55, 59, 135, 139] [57, 61, 74, 78, 137, 141] [7, 59, 69, 139, 149] [8, 22, 60, 70, 140, 150] [61, 67, 71, 76, 141, 151] [11, 26, 62, 66, 131, 142] [55, 64, 68, 69, 133, 144] [4, 15, 66, 73, 142, 146] [0, 3, 67, 74, 147, 158] [43, 59, 68, 75, 144, 148] [18, 34, 70, 72, 150, 152] [8, 22, 72, 77, 152, 157] [1, 10, 74, 76, 147, 154] [0, 3, 76, 78, 154, 156]
Code ID 159-40-3 · download JSON · raw on GitHub