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[[192,4,19]] d ≤
n
192
k
4
d
19
kd²/n
7.521
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)
witness operator (support, 19 qubits)
[7, 10, 31, 36, 41, 43, 53, 58, 64, 79, 82, 99, 109, 110, 117, 119, 120, 132, 149]
d_Z 19 · witness weight 19 (claimed upper_bound)
witness operator (support, 19 qubits)
[12, 16, 20, 23, 32, 37, 46, 56, 77, 90, 109, 112, 121, 127, 152, 155, 171, 173, 186]
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)×96 (2,4)×288 (3,4)×176 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 96 (1,4): 96 (2,4): 288 (2,5): 1152 (2,6): 576 (3,4): 176 (3,5): 1632 (3,6): 9456 (3,7): 13632 (3,8): 5760 (3,9): 1728 (3,10): 384
trapping sets H_Z (1,3)×96 (2,4)×288 (3,4)×176 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 96 (1,4): 96 (2,4): 288 (2,5): 1152 (2,6): 576 (3,4): 176 (3,5): 1632 (3,6): 9456 (3,7): 13632 (3,8): 5760 (3,9): 1728 (3,10): 384

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

[[192,4,19]] — two-block group-algebra code on SmallGroup(96,1)

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. Efficiency kd²/n = 4·361/192 = 7.52.

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_order96_k4.txt, line 7. This row is SmallGroup(96,1) with nonidentity GAP supports a=[2,16], b=[8,11,57]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 19, not an exact certificate. The published distance (d=19) 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(96,1), 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=[2,16] and b=[8,11,57] 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) = 4. 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 7) · Z-checks 96 (max weight 7)
H_X (96 checks, sparse supports)
[0, 73, 94, 96, 144, 149, 151] [0, 1, 89, 97, 101, 139, 158] [1, 2, 22, 98, 106, 153, 169] [3, 7, 40, 97, 99, 121, 171] [4, 23, 65, 100, 104, 145, 173] [5, 48, 85, 101, 122, 123, 169] [6, 62, 89, 102, 121, 186, 189] [2, 7, 11, 96, 103, 110, 160] [3, 8, 26, 98, 104, 113, 141] [4, 9, 50, 105, 108, 115, 165] [5, 10, 74, 96, 106, 114, 130] [6, 11, 95, 107, 113, 143, 190] [8, 12, 44, 103, 108, 124, 155] [9, 13, 46, 109, 119, 126, 176] [10, 14, 47, 97, 110, 125, 144] [0, 11, 15, 111, 124, 157, 191] [16, 24, 67, 105, 112, 147, 185] [17, 25, 68, 103, 106, 113, 146] [2, 18, 26, 114, 129, 145, 175] [19, 49, 86, 115, 119, 124, 148] [12, 20, 50, 116, 147, 154, 187] [21, 34, 74, 103, 117, 177, 191] [22, 84, 95, 118, 170, 175, 176] [12, 23, 29, 99, 119, 131, 162] [13, 24, 31, 100, 120, 133, 180] [14, 25, 32, 98, 101, 121, 132] [15, 26, 33, 118, 122, 131, 164] [16, 27, 52, 102, 109, 123, 137] [17, 28, 53, 99, 110, 116, 124] [18, 29, 54, 115, 125, 136, 167] [19, 30, 75, 100, 126, 131, 138] [20, 31, 76, 127, 137, 161, 168] [21, 32, 90, 99, 118, 128, 181] [22, 33, 73, 117, 129, 167, 180] [27, 34, 70, 107, 120, 130, 151] [28, 35, 71, 104, 121, 127, 131] [3, 29, 36, 126, 132, 150, 178] [30, 37, 72, 105, 133, 145, 152] [4, 31, 38, 134, 151, 172, 179] [5, 32, 39, 104, 129, 135, 186] [6, 33, 40, 128, 136, 178, 185] [41, 51, 87, 120, 126, 137, 170] [13, 42, 52, 138, 156, 169, 188] [14, 43, 53, 119, 139, 150, 157] [15, 44, 54, 107, 140, 149, 187] [35, 45, 75, 120, 141, 172, 178] [36, 46, 76, 125, 142, 171, 188] [47, 63, 90, 143, 150, 152, 185] [34, 48, 56, 111, 112, 144, 158] [35, 49, 57, 108, 113, 134, 145] [36, 50, 58, 133, 140, 146, 182] [37, 51, 59, 109, 115, 147, 159] [38, 52, 60, 142, 148, 158, 183] [39, 53, 61, 108, 136, 143, 149] [1, 40, 54, 102, 135, 150, 182] [41, 55, 77, 112, 117, 133, 151] [42, 56, 78, 96, 152, 163, 184] [43, 57, 79, 100, 140, 153, 164] [7, 44, 58, 111, 139, 154, 168] [45, 59, 91, 112, 142, 155, 182] [23, 46, 60, 132, 141, 156, 184] [47, 48, 61, 102, 140, 157, 159] [55, 62, 88, 123, 128, 147, 158] [16, 56, 63, 97, 159, 174, 189] [17, 57, 64, 105, 154, 160, 175] [18, 58, 65, 122, 153, 161, 179] [19, 59, 66, 123, 156, 162, 187] [20, 60, 67, 146, 155, 163, 189] [21, 61, 68, 107, 154, 164, 170] [37, 69, 77, 144, 165, 174, 179] [38, 70, 78, 127, 166, 173, 191] [39, 71, 79, 122, 128, 167, 172] [64, 72, 91, 146, 153, 168, 174] [62, 73, 80, 130, 135, 137, 169] [63, 74, 81, 98, 166, 170, 190] [64, 75, 82, 109, 161, 167, 171] [8, 65, 76, 114, 160, 172, 183] [66, 77, 83, 130, 163, 168, 173] [9, 67, 78, 116, 162, 174, 190] [10, 68, 79, 111, 117, 161, 175] [69, 80, 92, 101, 166, 176, 183] [24, 70, 81, 118, 134, 165, 177] [25, 71, 82, 114, 135, 142, 178] [49, 72, 83, 116, 160, 166, 179] [41, 80, 84, 106, 177, 180, 188] [42, 81, 85, 129, 148, 176, 181] [43, 82, 86, 125, 149, 156, 182] [45, 83, 87, 127, 171, 177, 183] [66, 88, 92, 148, 155, 184, 186] [84, 89, 93, 110, 181, 184, 185] [27, 85, 90, 136, 138, 180, 186] [28, 86, 91, 132, 139, 163, 187] [30, 87, 92, 134, 141, 181, 188] [51, 88, 93, 138, 143, 162, 189] [69, 93, 94, 152, 157, 173, 190] [55, 94, 95, 159, 164, 165, 191]
H_Z (96 checks, sparse supports)
[0, 7, 10, 56, 96, 97, 111] [1, 3, 14, 63, 97, 98, 150] [2, 8, 25, 74, 98, 103, 114] [3, 23, 28, 32, 99, 104, 132] [4, 24, 30, 57, 100, 105, 134] [1, 5, 25, 80, 101, 106, 135] [6, 27, 54, 61, 102, 107, 136] [7, 12, 17, 21, 99, 103, 154] [4, 8, 35, 39, 104, 108, 172] [9, 16, 37, 64, 105, 109, 174] [2, 10, 17, 84, 106, 110, 175] [11, 34, 44, 68, 103, 107, 111] [9, 12, 49, 53, 108, 116, 119] [13, 27, 51, 75, 109, 120, 138] [7, 14, 28, 89, 110, 121, 139] [15, 48, 58, 79, 111, 122, 140] [16, 48, 55, 59, 112, 123, 159] [8, 11, 17, 49, 113, 124, 160] [10, 18, 76, 82, 114, 125, 161] [9, 19, 29, 51, 115, 126, 162] [20, 28, 78, 83, 116, 127, 163] [21, 33, 55, 79, 117, 128, 164] [22, 26, 32, 81, 98, 118, 129] [13, 19, 23, 43, 100, 119, 156] [24, 34, 41, 45, 112, 120, 177] [3, 6, 25, 35, 113, 121, 178] [5, 26, 65, 71, 104, 114, 122] [5, 27, 62, 66, 123, 130, 186] [12, 15, 19, 28, 124, 131, 187] [14, 29, 46, 86, 119, 125, 132] [13, 30, 36, 41, 126, 133, 188] [31, 35, 70, 87, 120, 127, 134] [32, 40, 62, 71, 121, 128, 135] [18, 33, 39, 85, 122, 129, 136] [10, 34, 73, 77, 117, 130, 144] [23, 26, 30, 35, 131, 141, 145] [25, 36, 60, 91, 132, 142, 146] [24, 37, 50, 55, 133, 147, 165] [38, 49, 81, 92, 134, 148, 166] [39, 54, 73, 82, 135, 149, 167] [29, 40, 53, 90, 99, 136, 150] [27, 31, 41, 73, 137, 151, 180] [30, 42, 90, 93, 138, 152, 181] [1, 43, 58, 91, 139, 153, 182] [44, 50, 57, 61, 108, 140, 154] [8, 45, 60, 92, 141, 155, 183] [46, 52, 59, 82, 109, 142, 156] [11, 47, 53, 93, 110, 143, 157] [0, 14, 48, 69, 101, 144, 157] [4, 18, 37, 49, 115, 145, 179] [17, 50, 67, 72, 105, 116, 146] [16, 20, 51, 62, 137, 147, 189] [19, 52, 85, 88, 123, 138, 148] [0, 44, 53, 86, 124, 139, 149] [36, 43, 47, 54, 125, 140, 150] [0, 34, 38, 55, 151, 158, 191] [37, 47, 56, 94, 144, 152, 159] [2, 57, 65, 72, 145, 153, 160] [20, 58, 64, 68, 146, 154, 161] [12, 59, 67, 88, 147, 155, 162] [42, 60, 66, 86, 148, 156, 163] [15, 43, 61, 94, 149, 157, 164] [1, 48, 52, 62, 102, 158, 169] [51, 61, 63, 95, 143, 159, 170] [7, 64, 76, 83, 160, 168, 171] [31, 65, 75, 79, 100, 161, 172] [23, 66, 78, 93, 162, 173, 184] [56, 67, 77, 91, 112, 163, 174] [26, 57, 68, 95, 113, 164, 175] [9, 69, 81, 95, 165, 176, 190] [70, 74, 80, 83, 130, 166, 177] [29, 33, 71, 75, 131, 167, 178] [31, 58, 72, 77, 133, 168, 179] [2, 5, 42, 73, 96, 129, 169] [22, 41, 68, 74, 106, 117, 170] [3, 46, 75, 87, 126, 141, 171] [38, 45, 71, 76, 127, 142, 172] [4, 70, 77, 94, 151, 165, 173] [63, 69, 72, 78, 152, 166, 174] [18, 22, 64, 79, 153, 167, 175] [13, 22, 80, 85, 169, 176, 180] [21, 81, 84, 87, 170, 177, 181] [36, 40, 45, 82, 171, 178, 182] [38, 65, 69, 83, 173, 179, 183] [24, 33, 84, 90, 118, 180, 185] [32, 85, 89, 92, 101, 181, 186] [50, 54, 59, 86, 115, 182, 187] [52, 76, 80, 87, 137, 183, 188] [56, 60, 88, 89, 158, 184, 189] [16, 40, 47, 89, 97, 102, 185] [6, 39, 88, 90, 128, 143, 186] [20, 44, 66, 91, 155, 168, 187] [42, 46, 84, 92, 176, 184, 188] [6, 63, 67, 93, 185, 189, 190] [11, 74, 78, 94, 96, 190, 191] [15, 21, 70, 95, 107, 118, 191]
Code ID 192-4-19 · download JSON · raw on GitHub