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[[192,16,12]] d ≤
n
192
k
16
d
12
kd²/n
12.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[8, 9, 11, 26, 76, 77, 81, 83, 108, 122, 136, 154]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[7, 56, 59, 64, 72, 84, 91, 115, 128, 144, 151, 154]
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 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×192 (2,4)×1440 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 192 (2,4): 1440 (3,3): 96 (3,5): 14112 (3,7): 1920
trapping sets H_Z (1,3)×192 (2,4)×1440 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 192 (2,4): 1440 (3,3): 96 (3,5): 14112 (3,7): 1920

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 ≤ 6 (computed)

How this code was found

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

[[192,16,12]] — two-block group-algebra code on SmallGroup(96,12)

Direction & hypothesis

Target: unrestricted weight-6 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. Efficiency kd²/n = 12.00.

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_wt6wtL3_order96_k16.txt, line 14. This row is SmallGroup(96,12) with nonidentity GAP supports a=[8, 30], b=[37, 95]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 12, not an exact certificate. The published distance (d=12) 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(96,12), 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=[8, 30] and b=[37, 95] 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 = 192 - rank(HX) - rank(HZ) = 16. 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 96 (max weight 6) · Z-checks 96 (max weight 6)
H_X (96 checks, sparse supports)
[0, 57, 73, 96, 134, 136] [1, 12, 44, 97, 148, 150] [2, 30, 75, 98, 138, 140] [3, 32, 51, 99, 159, 161] [4, 23, 80, 100, 135, 163] [5, 29, 48, 101, 162, 164] [6, 82, 89, 98, 102, 163] [0, 7, 65, 103, 152, 154] [2, 8, 22, 104, 170, 172] [9, 34, 70, 105, 149, 174] [10, 35, 71, 106, 173, 175] [3, 11, 36, 103, 107, 174] [12, 53, 55, 108, 116, 118] [1, 13, 89, 109, 139, 166] [9, 14, 50, 110, 165, 167] [15, 59, 91, 99, 111, 166] [7, 16, 59, 112, 160, 181] [11, 17, 24, 113, 180, 182] [18, 61, 77, 108, 114, 181] [19, 49, 56, 111, 115, 183] [20, 50, 93, 109, 116, 164] [21, 58, 74, 110, 117, 183] [22, 28, 95, 109, 111, 118] [3, 23, 40, 119, 127, 129] [4, 24, 85, 120, 153, 177] [5, 25, 86, 121, 176, 178] [6, 12, 26, 104, 122, 177] [13, 27, 46, 123, 171, 186] [14, 28, 47, 124, 185, 187] [0, 15, 29, 119, 125, 186] [30, 62, 88, 122, 126, 188] [16, 31, 63, 120, 127, 175] [17, 32, 64, 121, 128, 188] [18, 33, 65, 120, 122, 129] [8, 34, 77, 117, 130, 142] [26, 27, 35, 131, 141, 143] [36, 79, 80, 96, 132, 142] [10, 37, 74, 114, 133, 184] [11, 38, 95, 112, 134, 167] [31, 39, 76, 113, 135, 184] [40, 49, 83, 112, 114, 136] [25, 31, 41, 132, 137, 190] [26, 42, 83, 130, 138, 182] [33, 43, 52, 131, 139, 190] [10, 44, 92, 130, 132, 140] [45, 75, 81, 133, 136, 141] [46, 55, 76, 110, 134, 142] [8, 47, 90, 133, 135, 143] [16, 48, 67, 128, 144, 156] [17, 49, 68, 145, 155, 157] [2, 18, 50, 97, 146, 156] [19, 51, 94, 125, 147, 189] [20, 34, 52, 123, 148, 178] [21, 35, 53, 124, 149, 189] [22, 36, 54, 123, 125, 150] [37, 55, 72, 146, 151, 191] [4, 38, 56, 144, 152, 187] [5, 39, 57, 145, 153, 191] [6, 40, 58, 144, 146, 154] [41, 59, 84, 147, 150, 155] [42, 60, 85, 121, 148, 156] [43, 61, 86, 147, 149, 157] [28, 52, 62, 102, 158, 168] [29, 63, 92, 100, 143, 159] [54, 56, 64, 101, 160, 168] [25, 65, 93, 100, 102, 161] [32, 66, 90, 137, 140, 162] [33, 67, 73, 113, 138, 163] [23, 60, 68, 137, 139, 164] [53, 60, 69, 158, 161, 165] [30, 54, 70, 131, 159, 166] [1, 71, 78, 158, 160, 167] [27, 72, 91, 98, 162, 168] [41, 73, 87, 107, 169, 179] [13, 42, 74, 105, 157, 170] [14, 43, 75, 106, 171, 179] [15, 44, 76, 105, 107, 172] [45, 62, 77, 151, 154, 173] [46, 63, 78, 124, 152, 174] [47, 64, 79, 151, 153, 175] [19, 66, 80, 169, 172, 176] [20, 67, 81, 145, 170, 177] [21, 68, 82, 169, 171, 178] [69, 83, 94, 103, 173, 179] [57, 78, 84, 115, 118, 180] [51, 58, 85, 101, 116, 181] [7, 81, 86, 115, 117, 182] [48, 61, 87, 99, 165, 183] [9, 79, 88, 108, 180, 184] [37, 69, 89, 126, 129, 185] [38, 70, 90, 106, 127, 186] [39, 71, 91, 126, 128, 187] [72, 84, 92, 104, 176, 188] [45, 87, 93, 119, 185, 189] [24, 82, 94, 96, 141, 190] [66, 88, 95, 97, 155, 191]
H_Z (96 checks, sparse supports)
[0, 36, 94, 96, 103, 125] [1, 50, 95, 97, 109, 167] [2, 6, 72, 98, 104, 146] [3, 15, 87, 99, 107, 119] [4, 63, 65, 100, 120, 152] [5, 64, 85, 101, 121, 153] [6, 62, 65, 102, 122, 154] [7, 11, 83, 103, 112, 182] [8, 26, 92, 104, 130, 143] [9, 74, 76, 105, 110, 184] [10, 75, 90, 106, 133, 140] [11, 73, 76, 107, 113, 134] [12, 18, 88, 97, 108, 122] [13, 20, 22, 109, 123, 170] [14, 21, 46, 110, 124, 171] [15, 19, 22, 111, 125, 172] [16, 38, 40, 112, 127, 144] [17, 39, 67, 113, 128, 145] [18, 37, 40, 114, 129, 146] [19, 84, 86, 115, 147, 176] [12, 20, 85, 116, 148, 177] [21, 34, 86, 117, 149, 178] [12, 22, 84, 104, 118, 150] [23, 29, 93, 100, 119, 164] [24, 31, 33, 113, 120, 190] [25, 32, 60, 121, 137, 161] [26, 30, 33, 122, 131, 138] [27, 52, 54, 123, 131, 168] [28, 53, 78, 118, 124, 158] [29, 51, 54, 101, 125, 159] [30, 89, 91, 98, 126, 166] [23, 31, 90, 127, 135, 137] [32, 48, 91, 99, 128, 162] [23, 33, 89, 129, 139, 163] [34, 42, 44, 105, 130, 148] [35, 43, 70, 106, 131, 149] [36, 41, 44, 107, 132, 150] [37, 45, 47, 133, 151, 185] [0, 38, 46, 134, 152, 186] [4, 39, 47, 135, 153, 187] [0, 40, 45, 119, 136, 154] [41, 66, 68, 137, 155, 169] [2, 42, 67, 138, 156, 170] [13, 43, 68, 139, 157, 171] [2, 44, 66, 97, 140, 172] [35, 45, 94, 141, 173, 189] [34, 36, 46, 123, 142, 174] [35, 47, 63, 124, 143, 175] [48, 56, 58, 101, 144, 183] [49, 57, 81, 115, 136, 145] [50, 55, 58, 110, 116, 146] [51, 59, 61, 99, 147, 181] [1, 52, 60, 139, 148, 158] [9, 53, 61, 108, 149, 165] [1, 54, 59, 150, 160, 166] [55, 77, 79, 108, 142, 151] [7, 56, 78, 115, 152, 160] [24, 57, 79, 96, 153, 180] [7, 58, 77, 117, 154, 181] [49, 59, 95, 111, 112, 155] [48, 50, 60, 156, 164, 165] [49, 61, 74, 114, 157, 183] [62, 69, 71, 126, 158, 173] [3, 63, 70, 127, 159, 174] [16, 64, 71, 128, 160, 175] [3, 65, 69, 103, 129, 161] [5, 66, 72, 162, 176, 191] [4, 6, 67, 144, 163, 177] [5, 20, 68, 145, 164, 178] [14, 69, 87, 165, 179, 185] [13, 15, 70, 105, 166, 186] [14, 38, 71, 106, 167, 187] [62, 64, 72, 151, 168, 188] [73, 80, 82, 96, 163, 169] [8, 74, 81, 117, 133, 170] [27, 75, 82, 98, 141, 171] [8, 76, 80, 135, 142, 172] [10, 77, 83, 114, 130, 173] [9, 11, 78, 167, 174, 180] [10, 31, 79, 132, 175, 184] [25, 80, 92, 100, 132, 176] [24, 26, 81, 141, 177, 182] [25, 52, 82, 102, 178, 190] [73, 75, 83, 136, 138, 179] [17, 84, 88, 155, 180, 188] [16, 18, 85, 120, 156, 181] [17, 42, 86, 121, 157, 182] [19, 21, 87, 169, 183, 189] [37, 39, 88, 126, 184, 191] [28, 89, 93, 102, 109, 185] [27, 29, 90, 143, 162, 186] [28, 56, 91, 111, 168, 187] [30, 32, 92, 140, 159, 188] [51, 53, 93, 116, 161, 189] [41, 43, 94, 147, 179, 190] [55, 57, 95, 118, 134, 191]
Code ID 192-16-12 · download JSON · raw on GitHub