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[[186,64,6]] d ≤
n
186
k
64
d
6
kd²/n
12.387
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[7, 49, 78, 125, 181, 185]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 31, 81, 107, 143, 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 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 2 · H_Z 2
trapping sets H_X (1,2)×186 (2,2)×930 (3,2)×4650 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 186 (2,2): 930 (3,2): 4650 (3,4): 1240
trapping sets H_Z (1,2)×186 (2,2)×930 (3,2)×4650 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 186 (2,2): 930 (3,2): 4650 (3,4): 1240

Circuit tier

syndrome-extraction memory circuits committed under circuits/186-64-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[22334, 22369, 22489]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[16967, 16971, 17616]
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 Construction I of arXiv:2610.19451, N=31
model Beam 1.0 (claimed, not verified)
date 2026-10-17
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

[[186,64,6]] — Construction I, N=31

Direction & hypothesis

Aimed at the CSS weight-6 / unrestricted cell using the constructions of arXiv:2610.19451 (Nourozi, Smarandache, Mitchell). Construction I is a 2x6 circulant family with analytic distance 6 and girth 12; Construction II is a mixed circulant-affine 2-block family with larger certified distances.

What was searched

The paper's fixed constructions were rebuilt from the construction described in arXiv:2610.19451 and screened against the board Pareto frontier. The undominated members listed above were submitted.

Evidence trail

Construction I carries the paper's analytic witnesses (Eq. (12) and its isomorphism mirror), so d <= 6 is a witnessed upper bound. Construction II (q=6) carries a witness found by a fast-RIS ladder; the paper certifies d = 20 by bounded enumeration but does not publish the operators.

Dead ends

Construction I with N = 63 and Construction II with q = 2, 3, 4 pass validation but are dominated by existing board entries and were not submitted.

Tools

Model: Beam 1.0. Repo tooling: verify/qldpc_verify.py.

Reproduction

Rebuild from arXiv:2610.19451: Construction I uses fixed circulant lists a=(0,1,3,8,12,18), b=(0,-5,-5,0,12,12), c=(0,0,2,8,2,8) mod N; Construction II uses the fixed polynomials A(z), B(z) over Z_24 with outer lift q=6.

Parity checks

X-checks 62 (max weight 6) · Z-checks 62 (max weight 6)
H_X (62 checks, sparse supports)
[0, 31, 62, 93, 124, 155] [1, 32, 63, 94, 125, 156] [2, 33, 64, 95, 126, 157] [3, 34, 65, 96, 127, 158] [4, 35, 66, 97, 128, 159] [5, 36, 67, 98, 129, 160] [6, 37, 68, 99, 130, 161] [7, 38, 69, 100, 131, 162] [8, 39, 70, 101, 132, 163] [9, 40, 71, 102, 133, 164] [10, 41, 72, 103, 134, 165] [11, 42, 73, 104, 135, 166] [12, 43, 74, 105, 136, 167] [13, 44, 75, 106, 137, 168] [14, 45, 76, 107, 138, 169] [15, 46, 77, 108, 139, 170] [16, 47, 78, 109, 140, 171] [17, 48, 79, 110, 141, 172] [18, 49, 80, 111, 142, 173] [19, 50, 81, 112, 143, 174] [20, 51, 82, 113, 144, 175] [21, 52, 83, 114, 145, 176] [22, 53, 84, 115, 146, 177] [23, 54, 85, 116, 147, 178] [24, 55, 86, 117, 148, 179] [25, 56, 87, 118, 149, 180] [26, 57, 88, 119, 150, 181] [27, 58, 89, 120, 151, 182] [28, 59, 90, 121, 152, 183] [29, 60, 91, 122, 153, 184] [30, 61, 92, 123, 154, 185] [0, 61, 90, 116, 143, 168] [1, 31, 91, 117, 144, 169] [2, 32, 92, 118, 145, 170] [3, 33, 62, 119, 146, 171] [4, 34, 63, 120, 147, 172] [5, 35, 64, 121, 148, 173] [6, 36, 65, 122, 149, 174] [7, 37, 66, 123, 150, 175] [8, 38, 67, 93, 151, 176] [9, 39, 68, 94, 152, 177] [10, 40, 69, 95, 153, 178] [11, 41, 70, 96, 154, 179] [12, 42, 71, 97, 124, 180] [13, 43, 72, 98, 125, 181] [14, 44, 73, 99, 126, 182] [15, 45, 74, 100, 127, 183] [16, 46, 75, 101, 128, 184] [17, 47, 76, 102, 129, 185] [18, 48, 77, 103, 130, 155] [19, 49, 78, 104, 131, 156] [20, 50, 79, 105, 132, 157] [21, 51, 80, 106, 133, 158] [22, 52, 81, 107, 134, 159] [23, 53, 82, 108, 135, 160] [24, 54, 83, 109, 136, 161] [25, 55, 84, 110, 137, 162] [26, 56, 85, 111, 138, 163] [27, 57, 86, 112, 139, 164] [28, 58, 87, 113, 140, 165] [29, 59, 88, 114, 141, 166] [30, 60, 89, 115, 142, 167]
H_Z (62 checks, sparse supports)
[0, 36, 67, 93, 143, 174] [1, 37, 68, 94, 144, 175] [2, 38, 69, 95, 145, 176] [3, 39, 70, 96, 146, 177] [4, 40, 71, 97, 147, 178] [5, 41, 72, 98, 148, 179] [6, 42, 73, 99, 149, 180] [7, 43, 74, 100, 150, 181] [8, 44, 75, 101, 151, 182] [9, 45, 76, 102, 152, 183] [10, 46, 77, 103, 153, 184] [11, 47, 78, 104, 154, 185] [12, 48, 79, 105, 124, 155] [13, 49, 80, 106, 125, 156] [14, 50, 81, 107, 126, 157] [15, 51, 82, 108, 127, 158] [16, 52, 83, 109, 128, 159] [17, 53, 84, 110, 129, 160] [18, 54, 85, 111, 130, 161] [19, 55, 86, 112, 131, 162] [20, 56, 87, 113, 132, 163] [21, 57, 88, 114, 133, 164] [22, 58, 89, 115, 134, 165] [23, 59, 90, 116, 135, 166] [24, 60, 91, 117, 136, 167] [25, 61, 92, 118, 137, 168] [26, 31, 62, 119, 138, 169] [27, 32, 63, 120, 139, 170] [28, 33, 64, 121, 140, 171] [29, 34, 65, 122, 141, 172] [30, 35, 66, 123, 142, 173] [0, 31, 91, 116, 153, 178] [1, 32, 92, 117, 154, 179] [2, 33, 62, 118, 124, 180] [3, 34, 63, 119, 125, 181] [4, 35, 64, 120, 126, 182] [5, 36, 65, 121, 127, 183] [6, 37, 66, 122, 128, 184] [7, 38, 67, 123, 129, 185] [8, 39, 68, 93, 130, 155] [9, 40, 69, 94, 131, 156] [10, 41, 70, 95, 132, 157] [11, 42, 71, 96, 133, 158] [12, 43, 72, 97, 134, 159] [13, 44, 73, 98, 135, 160] [14, 45, 74, 99, 136, 161] [15, 46, 75, 100, 137, 162] [16, 47, 76, 101, 138, 163] [17, 48, 77, 102, 139, 164] [18, 49, 78, 103, 140, 165] [19, 50, 79, 104, 141, 166] [20, 51, 80, 105, 142, 167] [21, 52, 81, 106, 143, 168] [22, 53, 82, 107, 144, 169] [23, 54, 83, 108, 145, 170] [24, 55, 84, 109, 146, 171] [25, 56, 85, 110, 147, 172] [26, 57, 86, 111, 148, 173] [27, 58, 87, 112, 149, 174] [28, 59, 88, 113, 150, 175] [29, 60, 89, 114, 151, 176] [30, 61, 90, 115, 152, 177]
Code ID 186-64-6 · download JSON · raw on GitHub