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[[186,10,16]] d ≤
n
186
k
10
d
16
kd²/n
13.763
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[7, 34, 42, 56, 65, 71, 100, 102, 108, 114, 135, 141, 158, 166, 171, 185]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 9, 15, 30, 53, 55, 68, 73, 84, 92, 118, 127, 162, 164, 173, 180]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 93 (1,4): 93 (2,4): 279 (2,5): 1116 (2,6): 558 (3,3): 93 (3,4): 62 (3,5): 1116 (3,6): 9455 (3,7): 13764 (3,8): 5673 (3,9): 1674 (3,10): 372
trapping sets H_Z (1,3)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 93 (1,4): 93 (2,4): 279 (2,5): 1116 (2,6): 558 (3,3): 93 (3,4): 62 (3,5): 1116 (3,6): 9455 (3,7): 13764 (3,8): 5673 (3,9): 1674 (3,10): 372

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). Checked against the live board: not equivalent to any existing entry.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[186,10,16]] — two-block group-algebra code on SmallGroup(93,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board (upstream main 72d7db93). Efficiency kd²/n = 10·256/186 = 13.76.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order93_k10.txt, line 2. This row is SmallGroup(93,1) with nonidentity GAP supports a=[3,36], b=[2,9,82]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=10), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 16, not an exact certificate. The published distance (d=16) is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

Union Alpha (maker currently anonymous), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(93,1), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[3,36] and b=[2,9,82] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 186 - rank(HX) - rank(HZ) = 10. Pass to the repository submission builder with confidence upper_bound; save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

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