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[[175,44,3]] d ≤
n
175
k
44
d
3
kd²/n
2.26286
w
6
X/Z
1.33
g
0.0057
r
4.4721
layers
2
swaps
247

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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)
[4, 12, 167, 173]
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)
[6, 161, 169]
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 5.939)
qubit degrees H_X 1–4 (mean 2.229) · H_Z 1–4 (mean 2.24)
trapping sets H_X (1,1)×45 (2,1)×93 (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): 45 (1,2): 65 (1,3): 45 (1,4): 20 (2,1): 93 (2,2): 200 (2,3): 285 (2,4): 213 (2,5): 101 (2,6): 19 (3,0): 9 (3,1): 396 (3,2): 930 (3,3): 1338 (3,4): 1548 (3,5): 1210 (3,6): 577 (3,7): 275 (3,8): 76
trapping sets H_Z (1,1)×44 (2,1)×88 (3,1)×392 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 44 (1,2): 66 (1,3): 44 (1,4): 21 (2,1): 88 (2,2): 197 (2,3): 282 (2,4): 214 (2,5): 105 (2,6): 20 (3,1): 392 (3,2): 889 (3,3): 1311 (3,4): 1538 (3,5): 1226 (3,6): 584 (3,7): 287 (3,8): 81
witness diameter X 1.4142 · Z 1.4142 (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/176-44-4.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 = 4.472
X checkZ checkqubit site (88)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 247 nearest-neighbor SWAPs per round in total, at most 6 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/176-44-4.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

[[175,44,3]] — priced calculus E-PEEL transform of codes/176-44-4.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/176-44-4.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/176-44-4.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/176-44-4.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/176-44-4.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 65 (max weight 6) · Z-checks 66 (max weight 6)
H_X (65 checks, sparse supports)
[0, 12, 87, 91, 165, 167] [1, 6, 89, 96, 161, 174] [2, 20, 88, 89, 90, 98] [3, 4, 87, 91, 172, 173] [0, 4, 92, 100, 165, 173] [5, 7, 87, 92, 93, 102] [6, 8, 88, 94, 169, 174] [7, 14, 87, 95, 100, 104] [9, 28, 93, 95, 97, 106] [10, 11, 90, 94, 96, 98] [2, 11, 88, 96, 99, 108] [13, 15, 90, 99, 101, 110] [14, 16, 91, 93, 100, 102] [15, 22, 90, 103, 108, 112] [17, 36, 101, 103, 105, 114] [18, 19, 97, 102, 104, 106] [9, 19, 93, 104, 107, 116] [21, 23, 97, 107, 109, 118] [22, 24, 98, 101, 108, 110] [23, 30, 97, 111, 116, 120] [25, 44, 109, 111, 113, 122] [26, 27, 105, 110, 112, 114] [17, 27, 101, 112, 115, 124] [29, 31, 105, 115, 117, 126] [30, 32, 106, 109, 116, 118] [31, 38, 105, 119, 124, 128] [33, 52, 117, 119, 121, 130] [34, 35, 113, 118, 120, 122] [25, 35, 109, 120, 123, 132] [37, 39, 113, 123, 125, 134] [38, 40, 114, 117, 124, 126] [39, 46, 113, 127, 132, 136] [41, 60, 125, 127, 129, 138] [42, 43, 121, 126, 128, 130] [33, 43, 117, 128, 131, 140] [45, 47, 121, 131, 133, 142] [46, 48, 122, 125, 132, 134] [47, 54, 121, 135, 140, 144] [49, 68, 133, 135, 137, 146] [50, 51, 129, 134, 136, 138] [41, 51, 125, 136, 139, 148] [53, 55, 129, 139, 141, 150] [54, 56, 130, 133, 140, 142] [55, 62, 129, 143, 148, 152] [57, 76, 141, 143, 145, 154] [58, 59, 137, 142, 144, 146] [49, 59, 133, 144, 147, 156] [61, 63, 137, 147, 149, 158] [62, 64, 138, 141, 148, 150] [63, 70, 137, 151, 156, 160] [65, 83, 149, 151, 153, 162] [66, 67, 145, 150, 152, 154] [57, 67, 141, 152, 155, 164] [69, 71, 145, 155, 157, 166] [70, 72, 146, 149, 156, 158] [71, 78, 145, 159, 164, 168] [73, 86, 157, 159, 161, 169] [74, 75, 153, 158, 160, 162] [65, 75, 149, 160, 163, 171] [77, 79, 153, 163, 165, 172] [78, 80, 154, 157, 164, 166] [79, 84, 153, 167, 171, 173] [81, 82, 161, 166, 168, 169] [73, 82, 157, 168, 170, 174] [84, 85, 162, 165, 171, 172]
H_Z (66 checks, sparse supports)
[0, 3, 5, 7, 87, 92] [2, 6, 11, 88, 96] [1, 2, 8, 20, 88, 89] [2, 10, 13, 15, 90, 99] [0, 3, 14, 16, 91, 100] [4, 5, 12, 16, 91, 92] [5, 9, 14, 19, 93, 104] [6, 10, 20, 89, 94] [7, 9, 16, 28, 93, 95] [9, 18, 21, 23, 97, 107] [2, 10, 22, 24, 98, 108] [11, 13, 20, 24, 98, 99] [13, 17, 22, 27, 101, 112] [5, 14, 18, 28, 95, 102] [15, 17, 24, 36, 101, 103] [17, 26, 29, 31, 105, 115] [9, 18, 30, 32, 106, 116] [19, 21, 28, 32, 106, 107] [21, 25, 30, 35, 109, 120] [13, 22, 26, 36, 103, 110] [23, 25, 32, 44, 109, 111] [25, 34, 37, 39, 113, 123] [17, 26, 38, 40, 114, 124] [27, 29, 36, 40, 114, 115] [29, 33, 38, 43, 117, 128] [21, 30, 34, 44, 111, 118] [31, 33, 40, 52, 117, 119] [33, 42, 45, 47, 121, 131] [25, 34, 46, 48, 122, 132] [35, 37, 44, 48, 122, 123] [37, 41, 46, 51, 125, 136] [29, 38, 42, 52, 119, 126] [39, 41, 48, 60, 125, 127] [41, 50, 53, 55, 129, 139] [33, 42, 54, 56, 130, 140] [43, 45, 52, 56, 130, 131] [45, 49, 54, 59, 133, 144] [37, 46, 50, 60, 127, 134] [47, 49, 56, 68, 133, 135] [49, 58, 61, 63, 137, 147] [41, 50, 62, 64, 138, 148] [51, 53, 60, 64, 138, 139] [53, 57, 62, 67, 141, 152] [45, 54, 58, 68, 135, 142] [55, 57, 64, 76, 141, 143] [57, 66, 69, 71, 145, 155] [49, 58, 70, 72, 146, 156] [59, 61, 68, 72, 146, 147] [61, 65, 70, 75, 149, 160] [53, 62, 66, 76, 143, 150] [63, 65, 72, 83, 149, 151] [65, 74, 77, 79, 153, 163] [57, 66, 78, 80, 154, 164] [67, 69, 76, 80, 154, 155] [69, 73, 78, 82, 157, 168] [61, 70, 74, 83, 151, 158] [71, 73, 80, 86, 157, 159] [1, 73, 81, 161, 170] [65, 74, 84, 85, 162, 171] [75, 77, 83, 85, 162, 163] [0, 4, 77, 84, 165, 173] [69, 78, 81, 86, 159, 166] [0, 12, 79, 85, 165, 167] [6, 8, 73, 81, 169, 174] [8, 82, 86, 169, 170] [3, 12, 77, 84, 167, 172]
Code ID 175-44-3 · download JSON · raw on GitHub