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[[192,12,14]] d ≤
n
192
k
12
d
14
kd²/n
12.25
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[10, 50, 95, 105, 115, 124, 134, 143, 148, 157, 160, 162, 173, 191]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[6, 7, 20, 31, 39, 71, 104, 109, 136, 141, 152, 163, 174, 189]
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 8 · H_Z 8 (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 (3,5)×14400 (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 (3,5): 14400 (3,7): 1920
trapping sets H_Z (1,3)×192 (2,4)×1440 (3,5)×14400 (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 (3,5): 14400 (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

[[192,12,14]] — two-block group-algebra code on SmallGroup(96,17)

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 = 12.25.

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_k12.txt, line 39. This row is SmallGroup(96,17) with nonidentity GAP supports a=[2, 16], b=[21, 86]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 14, not an exact certificate. The published distance (d=14) 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(96,17), 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=[2, 16] and b=[21, 86] 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) = 12. 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, 10, 65, 96, 134, 168] [0, 1, 74, 97, 148, 179] [2, 22, 25, 98, 138, 183] [3, 28, 40, 99, 159, 184] [4, 30, 85, 100, 111, 118] [1, 5, 86, 101, 142, 162] [6, 12, 32, 102, 115, 163] [2, 7, 59, 103, 152, 188] [3, 8, 52, 104, 170, 189] [4, 9, 75, 105, 122, 129] [5, 10, 89, 106, 156, 173] [6, 11, 90, 107, 126, 174] [12, 44, 49, 108, 116, 190] [13, 46, 51, 109, 114, 136] [7, 14, 47, 110, 163, 165] [0, 15, 53, 111, 133, 166] [16, 55, 67, 112, 132, 140] [8, 17, 68, 113, 166, 180] [2, 18, 57, 114, 137, 181] [9, 19, 94, 115, 135, 143] [20, 34, 59, 96, 116, 136] [11, 21, 35, 100, 117, 183] [22, 36, 61, 109, 118, 141] [12, 23, 80, 119, 127, 191] [11, 13, 24, 120, 125, 150] [14, 25, 31, 121, 174, 176] [15, 26, 83, 122, 147, 177] [16, 27, 53, 123, 146, 154] [17, 28, 77, 124, 177, 185] [18, 29, 78, 125, 151, 186] [19, 30, 50, 126, 149, 157] [20, 31, 91, 97, 127, 150] [21, 32, 95, 105, 128, 188] [22, 33, 48, 120, 129, 155] [34, 70, 73, 102, 130, 161] [23, 35, 71, 131, 141, 181] [3, 36, 75, 132, 142, 158] [24, 37, 72, 133, 139, 164] [4, 38, 77, 98, 134, 140] [5, 26, 39, 109, 135, 184] [6, 40, 79, 112, 136, 162] [27, 41, 87, 137, 160, 167] [13, 42, 80, 99, 138, 161] [14, 29, 43, 112, 139, 190] [15, 44, 82, 130, 140, 165] [31, 45, 62, 101, 141, 164] [46, 63, 83, 98, 102, 142] [33, 47, 64, 116, 133, 143] [29, 34, 48, 107, 144, 172] [35, 49, 56, 145, 155, 186] [36, 50, 93, 146, 156, 169] [32, 37, 51, 147, 153, 175] [33, 38, 52, 103, 148, 154] [39, 53, 60, 120, 149, 189] [30, 40, 54, 123, 150, 173] [26, 41, 55, 151, 171, 178] [42, 56, 79, 104, 152, 172] [43, 57, 92, 123, 153, 191] [24, 44, 58, 144, 154, 176] [45, 59, 76, 106, 155, 175] [46, 49, 60, 103, 107, 156] [47, 61, 73, 127, 147, 157] [48, 62, 88, 117, 158, 182] [16, 63, 89, 108, 118, 159] [17, 50, 64, 130, 160, 168] [18, 65, 91, 100, 161, 180] [19, 52, 66, 110, 162, 167] [20, 67, 92, 99, 111, 163] [21, 54, 68, 134, 137, 164] [37, 56, 69, 113, 165, 182] [38, 70, 93, 108, 114, 166] [39, 58, 71, 138, 158, 167] [60, 72, 84, 110, 117, 168] [57, 62, 73, 128, 169, 187] [58, 63, 74, 119, 129, 170] [64, 75, 81, 144, 171, 179] [55, 65, 76, 105, 172, 185] [61, 66, 77, 121, 173, 178] [1, 67, 78, 104, 122, 174] [9, 68, 79, 148, 151, 175] [54, 69, 80, 124, 176, 187] [25, 70, 81, 119, 125, 177] [51, 71, 82, 152, 169, 178] [23, 72, 83, 121, 128, 179] [41, 74, 84, 131, 143, 180] [42, 85, 95, 96, 132, 181] [43, 76, 86, 115, 159, 182] [45, 78, 87, 113, 135, 183] [66, 81, 88, 131, 139, 184] [82, 84, 89, 145, 157, 185] [8, 85, 90, 97, 146, 186] [27, 86, 91, 126, 170, 187] [10, 87, 92, 124, 149, 188] [7, 88, 93, 145, 153, 189] [69, 90, 94, 101, 160, 190] [28, 94, 95, 106, 171, 191]
H_Z (96 checks, sparse supports)
[0, 20, 85, 96, 97, 111] [1, 31, 90, 97, 101, 174] [2, 38, 46, 98, 103, 114] [3, 42, 67, 99, 104, 132] [4, 21, 65, 100, 105, 134] [5, 45, 94, 101, 106, 135] [6, 34, 46, 102, 107, 136] [7, 52, 60, 103, 110, 189] [8, 56, 78, 104, 113, 186] [9, 32, 76, 105, 115, 175] [10, 59, 95, 96, 106, 188] [11, 48, 60, 107, 117, 120] [12, 63, 70, 102, 108, 119] [13, 22, 39, 109, 120, 138] [14, 66, 72, 110, 121, 139] [4, 15, 67, 111, 122, 140] [16, 40, 43, 112, 123, 159] [17, 69, 87, 113, 124, 160] [13, 18, 70, 114, 125, 161] [6, 19, 86, 115, 126, 162] [12, 20, 47, 116, 127, 163] [21, 62, 72, 117, 128, 164] [4, 22, 63, 98, 118, 129] [23, 74, 81, 119, 131, 179] [24, 33, 53, 120, 133, 154] [25, 77, 83, 98, 121, 177] [9, 26, 78, 122, 135, 151] [27, 54, 57, 123, 137, 187] [28, 80, 92, 99, 124, 191] [24, 29, 81, 125, 139, 144] [11, 30, 91, 100, 126, 150] [23, 31, 61, 121, 127, 141] [32, 73, 83, 102, 128, 147] [9, 33, 74, 129, 143, 148] [34, 44, 64, 116, 130, 144] [35, 84, 88, 117, 131, 145] [16, 36, 85, 118, 132, 146] [15, 37, 47, 133, 147, 165] [0, 38, 68, 134, 148, 166] [19, 39, 87, 135, 149, 167] [13, 20, 40, 99, 136, 150] [18, 41, 68, 137, 151, 180] [2, 42, 71, 138, 152, 181] [37, 43, 88, 139, 153, 182] [16, 38, 44, 108, 140, 154] [22, 35, 45, 141, 155, 183] [5, 36, 46, 109, 142, 156] [19, 47, 84, 110, 143, 157] [48, 58, 75, 129, 144, 158] [49, 89, 93, 108, 145, 156] [27, 50, 90, 126, 146, 160] [26, 51, 61, 109, 147, 178] [1, 52, 79, 104, 148, 162] [30, 53, 92, 111, 123, 149] [24, 31, 54, 150, 164, 176] [29, 55, 79, 112, 151, 172] [7, 56, 82, 145, 152, 165] [51, 57, 93, 114, 153, 169] [27, 52, 58, 154, 167, 170] [33, 49, 59, 103, 116, 155] [10, 50, 60, 149, 156, 168] [30, 61, 89, 118, 157, 173] [36, 62, 71, 141, 158, 169] [3, 63, 86, 142, 159, 170] [41, 64, 94, 143, 160, 171] [34, 42, 65, 96, 161, 172] [5, 40, 66, 162, 173, 184] [6, 14, 67, 112, 163, 174] [37, 45, 68, 113, 164, 175] [14, 44, 69, 165, 176, 190] [15, 17, 70, 130, 166, 177] [41, 66, 71, 131, 167, 178] [0, 64, 72, 133, 168, 179] [50, 73, 82, 130, 157, 169] [8, 74, 91, 97, 170, 180] [55, 75, 95, 105, 132, 171] [48, 56, 76, 155, 172, 182] [10, 54, 77, 124, 134, 173] [11, 25, 78, 125, 174, 183] [51, 59, 79, 136, 152, 175] [25, 58, 80, 119, 138, 176] [26, 28, 81, 171, 177, 184] [55, 77, 82, 140, 178, 185] [1, 75, 83, 122, 142, 179] [17, 65, 84, 168, 180, 185] [18, 35, 85, 100, 181, 186] [62, 69, 86, 101, 182, 187] [2, 21, 87, 137, 183, 188] [3, 39, 88, 158, 184, 189] [28, 76, 89, 106, 159, 185] [29, 49, 90, 107, 186, 190] [73, 80, 91, 127, 161, 187] [7, 32, 92, 153, 163, 188] [8, 53, 93, 146, 166, 189] [12, 43, 94, 115, 190, 191] [23, 57, 95, 128, 181, 191]
Code ID 192-12-14 · download JSON · raw on GitHub