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[[167,10,13]] d ≤
n
167
k
10
d
13
kd²/n
10.1198
w
6
X/Z
1.08
g
0.0074
r
6.0828
layers
2
swaps
765

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Distance

X/Z asymmetry 1.08 · d_X ≤ 14, d_Z ≤ 13 · 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 14 · witness weight 14 (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, 14 qubits)
[6, 11, 15, 25, 32, 48, 52, 67, 73, 77, 82, 131, 132, 157]
d_Z 13 · witness weight 13 (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, 13 qubits)
[1, 8, 9, 15, 46, 53, 60, 74, 91, 92, 98, 138, 163]
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 5.962)
qubit degrees H_X 2–3 (mean 2.802) · H_Z 1–3 (mean 2.82)
trapping sets H_X (1,2)×33 (2,1)×1 (3,1)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 33 (1,3): 134 (2,1): 1 (2,2): 39 (2,3): 290 (2,4): 819 (3,1): 3 (3,2): 90 (3,3): 613 (3,4): 2922 (3,5): 6682 (3,6): 497 (3,7): 821
trapping sets H_Z (1,1)×1 (2,1)×1 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 1 (1,2): 28 (1,3): 138 (2,1): 1 (2,2): 38 (2,3): 250 (2,4): 860 (3,1): 1 (3,2): 80 (3,3): 548 (3,4): 2596 (3,5): 7107 (3,6): 434 (3,7): 886
witness diameter X 8.6023 · Z 9.0554 (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/168-10-14.json: the parent's coordinates with the entry of the removed qubit (parent index 73) 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 (86)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 765 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/168-10-14.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

[[167,10,13]] — priced calculus E-PEEL transform of codes/168-10-14.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/168-10-14.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/168-10-14.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/168-10-14.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/168-10-14.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 78 (max weight 6) · Z-checks 79 (max weight 6)
H_X (78 checks, sparse supports)
[0, 13, 34, 83, 117, 125] [1, 8, 35, 84, 120, 126] [2, 6, 27, 85, 110, 127] [3, 19, 39, 86, 122, 128] [4, 11, 32, 87, 115, 129] [5, 10, 37, 88, 113, 130] [6, 14, 40, 89, 124, 131] [3, 7, 30, 90, 118, 132] [8, 12, 33, 91, 116, 133] [4, 9, 25, 87, 92, 134] [10, 17, 38, 93, 121, 135] [11, 16, 41, 94, 119, 136] [5, 12, 20, 90, 95, 137] [9, 13, 36, 96, 123, 138] [0, 14, 18, 83, 97, 139] [10, 15, 31, 93, 98, 140] [2, 16, 23, 85, 99, 141] [7, 17, 22, 84, 100, 142] [11, 18, 26, 96, 101, 143] [1, 15, 19, 88, 102, 144] [3, 20, 24, 86, 103, 145] [16, 21, 37, 99, 104, 146] [8, 22, 29, 91, 105, 147] [13, 23, 28, 89, 106, 148] [17, 24, 32, 102, 107, 149] [6, 21, 25, 94, 108, 150] [9, 26, 30, 92, 109, 151] [22, 27, 41, 105, 110, 152] [14, 28, 35, 97, 111, 153] [19, 29, 34, 95, 112, 154] [23, 30, 38, 108, 113, 155] [12, 27, 31, 100, 114, 156] [15, 32, 36, 98, 115, 157] [7, 28, 33, 111, 116, 158] [20, 34, 40, 103, 117, 159] [25, 35, 39, 101, 118, 160] [2, 29, 36, 114, 119, 161] [18, 33, 37, 106, 120, 162] [1, 21, 38, 104, 121, 163] [5, 26, 39, 109, 122, 164] [4, 31, 40, 107, 123, 165] [0, 24, 41, 112, 124, 166] [42, 55, 75, 83, 127, 146] [43, 50, 76, 84, 130, 149] [44, 48, 69, 85, 129, 151] [45, 61, 80, 86, 133, 152] [46, 53, 73, 87, 125, 153] [47, 52, 78, 88, 132, 154] [48, 56, 81, 89, 136, 155] [45, 49, 72, 90, 126, 156] [50, 54, 74, 91, 135, 157] [46, 51, 67, 92, 139, 158] [52, 59, 79, 93, 128, 159] [53, 58, 82, 94, 138, 160] [47, 54, 62, 95, 142, 161] [51, 55, 77, 96, 131, 162] [42, 56, 60, 97, 141, 163] [44, 58, 65, 99, 134, 164] [49, 59, 64, 100, 144, 165] [53, 60, 68, 101, 126, 148] [43, 57, 61, 102, 137, 166] [45, 62, 66, 103, 127, 147] [58, 63, 78, 104, 128, 151] [50, 64, 71, 105, 129, 140] [55, 65, 70, 106, 130, 150] [59, 66, 73, 107, 131, 154] [48, 63, 67, 108, 132, 143] [51, 68, 72, 109, 133, 153] [64, 69, 82, 110, 134, 157] [56, 70, 76, 111, 135, 146] [61, 71, 75, 112, 136, 156] [65, 72, 79, 113, 137, 160] [57, 73, 77, 115, 139, 159] [49, 70, 74, 116, 140, 163] [62, 75, 81, 117, 141, 152] [67, 76, 80, 118, 142, 162] [43, 63, 79, 121, 145, 164] [47, 68, 80, 122, 147, 158]
H_Z (79 checks, sparse supports)
[0, 14, 42, 83, 97, 124] [1, 17, 43, 84, 102, 121] [2, 16, 44, 85, 99, 119] [3, 20, 45, 86, 90, 103] [4, 9, 46, 87, 92, 123] [5, 19, 47, 88, 95, 122] [6, 23, 48, 85, 89, 108] [7, 12, 49, 90, 100, 116] [8, 22, 50, 84, 91, 105] [9, 26, 51, 92, 96, 109] [10, 15, 52, 88, 93, 98] [11, 25, 53, 87, 94, 101] [12, 29, 54, 91, 95, 114] [13, 18, 55, 83, 96, 106] [14, 28, 56, 89, 97, 111] [15, 32, 57, 98, 102, 115] [16, 21, 58, 94, 99, 104] [17, 31, 59, 93, 100, 107] [18, 35, 60, 97, 101, 120] [19, 24, 61, 86, 102, 112] [20, 34, 62, 95, 103, 117] [21, 38, 63, 104, 108, 121] [22, 27, 64, 100, 105, 110] [23, 37, 65, 99, 106, 113] [24, 40, 66, 103, 107, 124] [25, 30, 67, 92, 108, 118] [26, 39, 68, 101, 109, 122] [2, 27, 69, 85, 110, 114] [28, 33, 70, 106, 111, 116] [29, 41, 71, 105, 112, 119] [5, 30, 72, 90, 109, 113] [31, 36, 98, 114, 123] [4, 32, 73, 87, 107, 115] [8, 33, 74, 91, 116, 120] [0, 34, 75, 83, 112, 117] [7, 35, 76, 84, 111, 118] [11, 36, 77, 96, 115, 119] [1, 37, 78, 88, 104, 120] [10, 38, 79, 93, 113, 121] [3, 39, 80, 86, 118, 122] [13, 40, 81, 89, 117, 123] [6, 41, 82, 94, 110, 124] [0, 46, 57, 125, 139, 166] [1, 49, 60, 126, 144, 163] [2, 42, 62, 127, 141, 161] [3, 52, 63, 128, 132, 145] [4, 44, 64, 129, 134, 165] [5, 43, 65, 130, 137, 164] [6, 55, 66, 127, 131, 150] [7, 47, 67, 132, 142, 158] [8, 45, 68, 126, 133, 147] [9, 58, 69, 134, 138, 151] [10, 50, 70, 130, 135, 140] [11, 48, 71, 129, 136, 143] [12, 61, 72, 133, 137, 156] [13, 53, 125, 138, 148] [14, 51, 73, 131, 139, 153] [15, 64, 74, 140, 144, 157] [16, 56, 75, 136, 141, 146] [17, 54, 76, 135, 142, 149] [18, 67, 77, 139, 143, 162] [19, 59, 78, 128, 144, 154] [20, 57, 79, 137, 145, 159] [21, 42, 70, 146, 150, 163] [22, 62, 80, 142, 147, 152] [23, 60, 81, 141, 148, 155] [24, 43, 145, 149, 166] [25, 65, 82, 134, 150, 160] [26, 44, 63, 143, 151, 164] [27, 45, 75, 127, 152, 156] [28, 46, 68, 148, 153, 158] [29, 47, 66, 147, 154, 161] [30, 48, 78, 132, 151, 155] [32, 50, 69, 129, 149, 157] [33, 51, 80, 133, 158, 162] [34, 52, 73, 125, 154, 159] [35, 53, 72, 126, 153, 160] [36, 54, 82, 138, 157, 161] [38, 56, 74, 135, 155, 163]
Code ID 167-10-13 · download JSON · raw on GitHub