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[[185,65,5]] d ≤
n
185
k
65
d
5
kd²/n
8.78378
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · 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 5 · witness weight 5 (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, 5 qubits)
[65, 89, 115, 152, 183]
d_Z 5 · witness weight 5 (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, 5 qubits)
[36, 122, 135, 160, 166]
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 12 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 1–2 (mean 1.946) · H_Z 1–2 (mean 1.946)
trapping sets H_X (1,1)×10 (2,1)×50 (3,1)×250 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 10 (1,2): 175 (2,1): 50 (2,2): 850 (3,1): 250 (3,2): 4125 (3,3): 100 (3,4): 1100
trapping sets H_Z (1,1)×10 (2,1)×50 (3,1)×250 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 10 (1,2): 175 (2,1): 50 (2,2): 850 (3,1): 250 (3,2): 4125 (3,3): 100 (3,4): 1100

Circuit tier

syndrome-extraction memory circuits committed under circuits/185-65-5/ · canonical noise recipe, 5 rounds, stim 1.16.0
d_circ ≤ 5 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[396, 424, 1276, 12849, 23157]
d_circ^Z 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[2243, 2311, 5336, 16752, 16767]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PUNCTURE transform of codes/186-64-6.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
family other (a tag, not a ranking)
locality unrestricted (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

[[185,65,5]] — priced calculus E-PUNCTURE transform of codes/186-64-6.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/186-64-6.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-PUNCTURE. Parent: codes/186-64-6.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/186-64-6.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/186-64-6.json by applying the E-PUNCTURE move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 60 (max weight 6) · Z-checks 60 (max weight 6)
H_X (60 checks, sparse supports)
[0, 30, 61, 92, 123, 154] [1, 31, 62, 93, 124, 155] [2, 32, 63, 94, 125, 156] [3, 33, 64, 95, 126, 157] [4, 34, 65, 96, 127, 158] [5, 35, 66, 97, 128, 159] [6, 36, 67, 98, 129, 160] [7, 37, 68, 99, 130, 161] [8, 38, 69, 100, 131, 162] [9, 39, 70, 101, 132, 163] [10, 40, 71, 102, 133, 164] [11, 41, 72, 103, 134, 165] [12, 42, 73, 104, 135, 166] [13, 43, 74, 105, 136, 167] [14, 44, 75, 106, 137, 168] [15, 45, 76, 107, 138, 169] [16, 46, 77, 108, 139, 170] [17, 47, 78, 109, 140, 171] [18, 48, 79, 110, 141, 172] [19, 49, 80, 111, 142, 173] [20, 50, 81, 112, 143, 174] [21, 51, 82, 113, 144, 175] [22, 52, 83, 114, 145, 176] [23, 53, 84, 115, 146, 177] [24, 54, 85, 116, 147, 178] [25, 55, 86, 117, 148, 179] [26, 56, 87, 118, 149, 180] [27, 57, 88, 119, 150, 181] [28, 58, 89, 120, 151, 182] [29, 59, 90, 121, 152, 183] [0, 60, 89, 115, 142, 167] [1, 30, 90, 116, 143, 168] [2, 31, 91, 117, 144, 169] [3, 32, 61, 118, 145, 170] [4, 33, 62, 119, 146, 171] [5, 34, 63, 120, 147, 172] [6, 35, 64, 121, 148, 173] [7, 36, 65, 122, 149, 174] [8, 37, 66, 92, 150, 175] [9, 38, 67, 93, 151, 176] [10, 39, 68, 94, 152, 177] [11, 40, 69, 95, 153, 178] [12, 41, 70, 96, 123, 179] [13, 42, 71, 97, 124, 180] [14, 43, 72, 98, 125, 181] [15, 44, 73, 99, 126, 182] [16, 45, 74, 100, 127, 183] [17, 46, 75, 101, 128, 184] [18, 47, 76, 102, 129, 154] [19, 48, 77, 103, 130, 155] [20, 49, 78, 104, 131, 156] [21, 50, 79, 105, 132, 157] [22, 51, 80, 106, 133, 158] [23, 52, 81, 107, 134, 159] [24, 53, 82, 108, 135, 160] [25, 54, 83, 109, 136, 161] [26, 55, 84, 110, 137, 162] [27, 56, 85, 111, 138, 163] [28, 57, 86, 112, 139, 164] [29, 58, 87, 113, 140, 165]
H_Z (60 checks, sparse supports)
[0, 35, 66, 92, 142, 173] [1, 36, 67, 93, 143, 174] [2, 37, 68, 94, 144, 175] [3, 38, 69, 95, 145, 176] [4, 39, 70, 96, 146, 177] [5, 40, 71, 97, 147, 178] [6, 41, 72, 98, 148, 179] [7, 42, 73, 99, 149, 180] [8, 43, 74, 100, 150, 181] [9, 44, 75, 101, 151, 182] [10, 45, 76, 102, 152, 183] [11, 46, 77, 103, 153, 184] [12, 47, 78, 104, 123, 154] [13, 48, 79, 105, 124, 155] [14, 49, 80, 106, 125, 156] [15, 50, 81, 107, 126, 157] [16, 51, 82, 108, 127, 158] [17, 52, 83, 109, 128, 159] [18, 53, 84, 110, 129, 160] [19, 54, 85, 111, 130, 161] [20, 55, 86, 112, 131, 162] [21, 56, 87, 113, 132, 163] [22, 57, 88, 114, 133, 164] [23, 58, 89, 115, 134, 165] [24, 59, 90, 116, 135, 166] [25, 60, 91, 117, 136, 167] [26, 30, 61, 118, 137, 168] [27, 31, 62, 119, 138, 169] [28, 32, 63, 120, 139, 170] [29, 33, 64, 121, 140, 171] [0, 30, 90, 115, 152, 177] [1, 31, 91, 116, 153, 178] [2, 32, 61, 117, 123, 179] [3, 33, 62, 118, 124, 180] [4, 34, 63, 119, 125, 181] [5, 35, 64, 120, 126, 182] [6, 36, 65, 121, 127, 183] [7, 37, 66, 122, 128, 184] [8, 38, 67, 92, 129, 154] [9, 39, 68, 93, 130, 155] [10, 40, 69, 94, 131, 156] [11, 41, 70, 95, 132, 157] [12, 42, 71, 96, 133, 158] [13, 43, 72, 97, 134, 159] [14, 44, 73, 98, 135, 160] [15, 45, 74, 99, 136, 161] [16, 46, 75, 100, 137, 162] [17, 47, 76, 101, 138, 163] [18, 48, 77, 102, 139, 164] [19, 49, 78, 103, 140, 165] [20, 50, 79, 104, 141, 166] [21, 51, 80, 105, 142, 167] [22, 52, 81, 106, 143, 168] [23, 53, 82, 107, 144, 169] [24, 54, 83, 108, 145, 170] [25, 55, 84, 109, 146, 171] [26, 56, 85, 110, 147, 172] [27, 57, 86, 111, 148, 173] [28, 58, 87, 112, 149, 174] [29, 59, 88, 113, 150, 175]
Code ID 185-65-5 · download JSON · raw on GitHub