[[181,6,17]] · QEC Challenge

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[[181,6,17]] d ≤

181

kd²/n

9.58011

X/Z

1.06

0.004

7.0

layers

swaps

1097

Share this result

Distance

X/Z asymmetry 1.06 · d_X ≤ 18, d_Z ≤ 17 · 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 18 · witness weight 18 (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, 18 qubits)

[5, 31, 33, 39, 67, 74, 88, 95, 98, 102, 124, 141, 147, 159, 164, 165, 169, 175]

d_Z 17 · witness weight 17 (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, 17 qubits)

[27, 37, 43, 47, 50, 66, 68, 82, 87, 89, 96, 117, 126, 135, 138, 145, 168]

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 5.977)

qubit degrees H_X 1–3 (mean 2.884) · H_Z 1–3 (mean 2.906)

trapping sets H_X (1,1)×1 (2,1)×1 (3,2)×26 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 1 (1,2): 19 (1,3): 161 (2,1): 1 (2,2): 7 (2,3): 183 (2,4): 1114 (3,2): 26 (3,3): 277 (3,4): 2083 (3,5): 9862 (3,6): 352 (3,7): 1366

trapping sets H_Z (1,1)×2 (2,2)×12 (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): 2 (1,2): 13 (1,3): 166 (2,2): 12 (2,3): 126 (2,4): 1172 (3,1): 1 (3,2): 16 (3,3): 294 (3,4): 1462 (3,5): 10600 (3,6): 244 (3,7): 1468

witness diameter X 11.4018 · Z 10.7703 (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/182-6-18.json: the parent's coordinates with the entry of the removed qubit (parent index 89) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04

X checkZ checkqubit site (93)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 1097 nearest-neighbor SWAPs per round in total, at most 13 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/182-6-18.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

[[181,6,17]] — priced calculus E-PEEL transform of codes/182-6-18.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/182-6-18.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/182-6-18.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/182-6-18.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/182-6-18.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 87 (max weight 6) · Z-checks 88 (max weight 6)

H_X (87 checks, sparse supports)

[0, 1, 18, 90, 92, 106] [1, 3, 24, 91, 94, 112] [2, 4, 25, 92, 95, 113] [3, 6, 31, 93, 97, 119] [4, 7, 32, 94, 98, 120] [5, 8, 33, 95, 99, 121] [6, 10, 38, 96, 101, 126] [7, 11, 39, 97, 102, 127] [8, 12, 40, 98, 103, 128] [9, 13, 41, 99, 104, 129] [10, 15, 45, 100, 106, 133] [11, 16, 46, 101, 107, 134] [12, 17, 47, 102, 108, 135] [13, 18, 48, 103, 109, 136] [3, 14, 19, 104, 110, 137] [15, 21, 52, 105, 112, 140] [16, 22, 53, 106, 113, 141] [17, 23, 54, 107, 114, 142] [18, 24, 55, 108, 115, 143] [6, 19, 25, 109, 116, 144] [7, 20, 26, 110, 117, 145] [21, 28, 59, 111, 119, 147] [22, 29, 60, 112, 120, 148] [23, 30, 61, 113, 121, 149] [24, 31, 62, 114, 122, 150] [10, 25, 32, 115, 123, 151] [11, 26, 33, 116, 124, 152] [12, 27, 34, 90, 100, 117] [28, 35, 66, 118, 126, 154] [29, 36, 67, 119, 127, 155] [30, 37, 68, 120, 128, 156] [31, 38, 69, 121, 129, 157] [15, 32, 39, 122, 130, 158] [16, 33, 40, 123, 131, 159] [17, 34, 41, 91, 105, 124] [35, 42, 72, 125, 133, 160] [36, 43, 73, 126, 134, 161] [37, 44, 74, 127, 135, 162] [38, 45, 75, 128, 136, 163] [21, 39, 46, 129, 137, 164] [22, 40, 47, 130, 138, 165] [23, 41, 48, 93, 111, 131] [42, 49, 77, 90, 132, 140] [43, 50, 78, 133, 141, 166] [44, 51, 79, 134, 142, 167] [45, 52, 80, 135, 143, 168] [28, 46, 53, 136, 144, 169] [29, 47, 54, 137, 145, 170] [30, 48, 55, 96, 118, 138] [49, 56, 81, 91, 139, 147] [50, 57, 82, 92, 140, 148] [51, 58, 83, 141, 149, 171] [52, 59, 84, 142, 150, 172] [35, 53, 60, 143, 151, 173] [36, 54, 61, 144, 152, 174] [37, 55, 62, 100, 125, 145] [5, 56, 63, 93, 146, 154] [57, 64, 85, 94, 147, 155] [58, 65, 86, 95, 148, 156] [59, 66, 87, 149, 157, 175] [42, 60, 67, 150, 158, 176] [43, 61, 68, 151, 159, 177] [44, 62, 69, 105, 132, 152] [8, 27, 63, 96, 153, 160] [9, 64, 70, 97, 154, 161] [65, 71, 88, 98, 155, 162] [49, 67, 73, 157, 164, 178] [50, 68, 74, 158, 165, 179] [51, 69, 75, 111, 139, 159] [0, 13, 70, 101, 160, 166] [14, 71, 76, 102, 161, 167] [72, 77, 89, 103, 162, 168] [56, 73, 78, 104, 163, 169] [57, 74, 79, 164, 170, 180] [58, 75, 80, 118, 146, 165] [2, 19, 76, 107, 166, 171] [20, 77, 81, 108, 167, 172] [63, 78, 82, 109, 168, 173] [64, 79, 83, 110, 169, 174] [65, 80, 84, 125, 153, 170] [5, 26, 81, 114, 171, 175] [27, 82, 85, 115, 172, 176] [70, 83, 86, 116, 173, 177] [71, 84, 87, 117, 132, 174] [9, 34, 85, 122, 175, 178] [0, 86, 88, 123, 176, 179] [1, 14, 88, 130, 178, 180]

H_Z (88 checks, sparse supports)

[0, 27, 42, 90, 160, 176] [1, 34, 49, 90, 91, 178] [0, 2, 50, 92, 166, 179] [3, 41, 56, 91, 93, 104] [1, 4, 57, 92, 94, 180] [2, 5, 58, 95, 146, 171] [6, 48, 63, 93, 96, 109] [3, 7, 64, 94, 97, 110] [4, 8, 65, 95, 98, 153] [5, 9, 66, 99, 154, 175] [10, 27, 55, 96, 100, 115] [6, 11, 70, 97, 101, 116] [7, 12, 71, 98, 102, 117] [8, 13, 72, 99, 103, 160] [9, 14, 73, 104, 161, 178] [15, 34, 62, 100, 105, 122] [0, 10, 16, 101, 106, 123] [11, 17, 76, 102, 107, 124] [12, 18, 77, 90, 103, 108] [13, 19, 78, 104, 109, 166] [14, 20, 79, 110, 167, 180] [21, 41, 69, 105, 111, 129] [1, 15, 22, 106, 112, 130] [2, 16, 23, 107, 113, 131] [17, 24, 81, 91, 108, 114] [18, 25, 82, 92, 109, 115] [19, 26, 83, 110, 116, 171] [20, 27, 84, 117, 153, 172] [28, 48, 75, 111, 118, 136] [3, 21, 29, 112, 119, 137] [4, 22, 30, 113, 120, 138] [5, 23, 31, 93, 114, 121] [24, 32, 85, 94, 115, 122] [25, 33, 86, 95, 116, 123] [26, 34, 87, 117, 124, 175] [35, 55, 80, 118, 125, 143] [6, 28, 36, 119, 126, 144] [7, 29, 37, 120, 127, 145] [8, 30, 38, 96, 121, 128] [9, 31, 39, 97, 122, 129] [32, 40, 88, 98, 123, 130] [33, 41, 99, 124, 131] [42, 62, 84, 125, 132, 150] [10, 35, 43, 126, 133, 151] [11, 36, 44, 127, 134, 152] [12, 37, 45, 100, 128, 135] [13, 38, 46, 101, 129, 136] [14, 39, 47, 102, 130, 137] [40, 48, 89, 103, 131, 138] [49, 69, 87, 132, 139, 157] [15, 42, 50, 133, 140, 158] [16, 43, 51, 134, 141, 159] [17, 44, 52, 105, 135, 142] [18, 45, 53, 106, 136, 143] [19, 46, 54, 107, 137, 144] [20, 47, 55, 108, 138, 145] [56, 75, 139, 146, 163] [21, 49, 57, 140, 147, 164] [22, 50, 58, 141, 148, 165] [23, 51, 59, 111, 142, 149] [24, 52, 60, 112, 143, 150] [25, 53, 61, 113, 144, 151] [26, 54, 62, 114, 145, 152] [63, 80, 89, 146, 153, 168] [28, 56, 64, 147, 154, 169] [29, 57, 65, 148, 155, 170] [30, 58, 66, 118, 149, 156] [31, 59, 67, 119, 150, 157] [32, 60, 68, 120, 151, 158] [33, 61, 69, 121, 152, 159] [35, 63, 70, 154, 160, 173] [36, 64, 71, 155, 161, 174] [37, 65, 72, 125, 156, 162] [38, 66, 73, 126, 157, 163] [39, 67, 74, 127, 158, 164] [40, 68, 75, 128, 159, 165] [43, 70, 76, 161, 166, 177] [44, 71, 77, 132, 162, 167] [45, 72, 78, 133, 163, 168] [46, 73, 79, 134, 164, 169] [47, 74, 80, 135, 165, 170] [51, 76, 81, 139, 167, 171] [52, 77, 82, 140, 168, 172] [53, 78, 83, 141, 169, 173] [54, 79, 84, 142, 170, 174] [59, 81, 85, 147, 172, 175] [60, 82, 86, 148, 173, 176] [67, 85, 88, 155, 176, 178]

Code ID 181-6-17 · download JSON · raw on GitHub