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[[200,8,18]] d ≤
n
200
k
8
d
18
kd²/n
12.96
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[29, 58, 63, 80, 82, 97, 101, 129, 133, 138, 139, 149, 164, 181, 182, 189, 193, 194]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[2, 4, 13, 15, 18, 41, 72, 89, 110, 123, 139, 151, 154, 168, 169, 182, 183, 198]
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 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 100 (1,5): 100 (2,2): 100 (2,5): 1000 (2,8): 1000 (3,2): 100 (3,5): 5000 (3,7): 500 (3,8): 18500 (3,9): 1800 (3,10): 2000 (3,11): 10600 (3,13): 1000
trapping sets H_Z (1,2)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 100 (1,5): 100 (2,2): 100 (2,5): 1000 (2,8): 1000 (3,2): 100 (3,5): 5000 (3,7): 500 (3,8): 18500 (3,9): 1800 (3,10): 2000 (3,11): 10600 (3,13): 1000

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

[[200,8,18]] — two-block group-algebra code on SmallGroup(100,6)

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 = 12.96.

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_wt7wtL2_order100_k8.txt, line 15. This row is SmallGroup(100,6) with nonidentity GAP supports a=[6], b=[5, 24, 39, 74]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, not an exact certificate. The published distance (d=18) 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(100,6), 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=[6] and b=[5, 24, 39, 74] 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 = 200 - rank(HX) - rank(HZ) = 8. 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 100 (max weight 7) · Z-checks 100 (max weight 7)
H_X (100 checks, sparse supports)
[0, 59, 100, 134, 139, 140, 169] [1, 32, 101, 143, 148, 149, 176] [2, 6, 102, 112, 151, 156, 157] [3, 41, 103, 124, 152, 158, 182] [4, 75, 100, 104, 153, 158, 183] [0, 5, 105, 117, 160, 165, 166] [6, 50, 106, 131, 161, 167, 187] [7, 51, 101, 107, 162, 167, 188] [8, 14, 104, 108, 122, 172, 173] [1, 9, 109, 123, 138, 168, 174] [10, 16, 102, 110, 124, 169, 174] [11, 60, 103, 111, 140, 170, 192] [12, 87, 103, 104, 112, 171, 193] [2, 13, 107, 113, 129, 179, 180] [3, 14, 114, 130, 147, 175, 181] [4, 15, 105, 115, 131, 176, 181] [16, 68, 106, 116, 149, 177, 195] [17, 69, 106, 107, 117, 178, 196] [18, 26, 110, 118, 136, 184, 185] [5, 19, 111, 119, 137, 155, 186] [20, 28, 108, 112, 120, 138, 186] [7, 21, 109, 121, 139, 157, 182] [22, 30, 109, 110, 122, 140, 183] [23, 76, 100, 111, 123, 184, 198] [24, 95, 111, 112, 124, 132, 185] [8, 25, 115, 125, 145, 189, 190] [9, 26, 116, 126, 146, 164, 191] [10, 27, 113, 117, 127, 147, 191] [11, 28, 114, 128, 148, 166, 187] [12, 29, 114, 115, 129, 149, 188] [30, 82, 101, 116, 130, 189, 199] [31, 83, 116, 117, 131, 141, 190] [32, 42, 120, 132, 153, 192, 193] [13, 33, 121, 133, 154, 171, 194] [34, 44, 118, 122, 134, 155, 194] [15, 35, 119, 123, 135, 156, 173] [36, 46, 119, 120, 124, 136, 157] [17, 37, 102, 121, 137, 158, 192] [38, 48, 100, 121, 122, 138, 193] [39, 88, 104, 123, 139, 150, 194] [40, 99, 118, 123, 124, 140, 151] [18, 41, 127, 141, 162, 195, 196] [19, 42, 128, 142, 163, 178, 197] [20, 43, 125, 129, 143, 164, 197] [21, 44, 126, 130, 144, 165, 180] [22, 45, 126, 127, 131, 145, 166] [23, 46, 105, 128, 146, 167, 195] [24, 47, 101, 128, 129, 147, 196] [48, 92, 107, 130, 148, 159, 197] [49, 93, 125, 130, 131, 149, 160] [25, 50, 135, 150, 170, 183, 198] [51, 61, 132, 136, 151, 171, 198] [27, 52, 133, 137, 152, 172, 185] [53, 63, 133, 134, 138, 153, 173] [29, 54, 108, 135, 139, 154, 174] [55, 65, 102, 135, 136, 140, 155] [31, 56, 103, 110, 137, 156, 198] [57, 67, 104, 132, 137, 138, 157] [58, 96, 112, 133, 139, 158, 168] [33, 59, 144, 159, 177, 188, 199] [34, 60, 141, 145, 160, 178, 199] [35, 61, 142, 146, 161, 179, 190] [36, 62, 142, 143, 147, 162, 180] [37, 63, 113, 144, 148, 163, 181] [38, 64, 105, 144, 145, 149, 164] [39, 65, 106, 115, 146, 165, 199] [40, 66, 107, 141, 146, 147, 166] [67, 98, 117, 142, 148, 167, 175] [43, 68, 150, 154, 168, 184, 193] [69, 77, 150, 151, 155, 169, 185] [45, 70, 118, 152, 156, 170, 186] [71, 79, 108, 152, 153, 157, 171] [47, 72, 109, 120, 154, 158, 172] [73, 81, 100, 110, 154, 155, 173] [49, 74, 111, 122, 150, 156, 174] [52, 75, 159, 163, 175, 189, 196] [53, 76, 159, 160, 164, 176, 190] [54, 77, 125, 161, 165, 177, 191] [55, 78, 113, 161, 162, 166, 178] [56, 79, 114, 127, 163, 167, 179] [57, 80, 101, 115, 163, 164, 180] [58, 81, 116, 129, 159, 165, 181] [62, 82, 132, 168, 172, 182, 194] [83, 89, 118, 168, 169, 173, 183] [64, 84, 119, 134, 170, 174, 184] [85, 91, 102, 120, 170, 171, 185] [66, 86, 103, 121, 136, 172, 186] [70, 87, 141, 175, 179, 187, 197] [71, 88, 125, 175, 176, 180, 188] [72, 89, 126, 143, 177, 181, 189] [73, 90, 105, 127, 177, 178, 190] [74, 91, 106, 128, 145, 179, 191] [78, 92, 133, 151, 182, 186, 192] [93, 97, 108, 134, 182, 183, 193] [80, 94, 109, 135, 153, 184, 194] [84, 95, 142, 160, 187, 191, 195] [85, 96, 113, 143, 187, 188, 196] [86, 97, 114, 144, 162, 189, 197] [90, 98, 119, 152, 169, 192, 198] [94, 99, 126, 161, 176, 195, 199]
H_Z (100 checks, sparse supports)
[0, 4, 23, 38, 73, 100, 105] [1, 7, 30, 47, 80, 101, 109] [2, 10, 37, 55, 85, 102, 113] [3, 11, 12, 56, 86, 103, 114] [4, 8, 12, 39, 57, 104, 115] [5, 15, 46, 64, 90, 105, 119] [6, 16, 17, 65, 91, 102, 106] [7, 13, 17, 48, 66, 107, 121] [8, 20, 54, 71, 93, 108, 125] [9, 21, 22, 72, 94, 109, 126] [10, 18, 22, 56, 73, 110, 127] [11, 19, 23, 24, 74, 111, 128] [2, 12, 20, 24, 58, 112, 129] [13, 27, 63, 78, 96, 113, 133] [14, 28, 29, 79, 97, 108, 114] [15, 25, 29, 65, 80, 115, 135] [16, 26, 30, 31, 81, 110, 116] [5, 17, 27, 31, 67, 117, 137] [18, 34, 40, 70, 83, 118, 141] [19, 35, 36, 84, 98, 119, 142] [20, 32, 36, 72, 85, 120, 143] [21, 33, 37, 38, 86, 121, 144] [8, 22, 34, 38, 74, 122, 145] [9, 23, 35, 39, 40, 123, 146] [3, 10, 24, 36, 40, 124, 147] [25, 43, 49, 77, 88, 125, 150] [26, 44, 45, 89, 99, 118, 126] [27, 41, 45, 79, 90, 127, 152] [28, 42, 46, 47, 91, 120, 128] [13, 29, 43, 47, 81, 129, 154] [14, 30, 44, 48, 49, 122, 130] [6, 15, 31, 45, 49, 131, 156] [24, 32, 51, 57, 82, 101, 132] [33, 52, 53, 58, 92, 133, 159] [0, 34, 53, 84, 93, 134, 160] [35, 50, 54, 55, 94, 135, 161] [18, 36, 51, 55, 86, 136, 162] [19, 37, 52, 56, 57, 137, 163] [9, 20, 38, 53, 57, 138, 164] [0, 21, 39, 54, 58, 139, 165] [0, 11, 22, 40, 55, 140, 166] [31, 41, 60, 66, 87, 103, 141] [42, 61, 62, 67, 95, 132, 142] [1, 43, 62, 89, 96, 143, 168] [44, 59, 63, 64, 97, 134, 144] [25, 45, 60, 64, 91, 145, 170] [26, 46, 61, 65, 66, 136, 146] [14, 27, 47, 62, 66, 147, 172] [1, 28, 48, 63, 67, 138, 148] [1, 16, 29, 49, 64, 149, 174] [39, 50, 68, 69, 74, 106, 150] [2, 40, 51, 69, 92, 107, 151] [3, 52, 70, 71, 98, 152, 175] [4, 32, 53, 71, 94, 153, 176] [33, 54, 68, 72, 73, 154, 177] [19, 34, 55, 69, 73, 155, 178] [2, 35, 56, 70, 74, 156, 179] [2, 21, 36, 57, 71, 157, 180] [3, 4, 37, 58, 72, 158, 181] [48, 59, 75, 76, 81, 100, 159] [5, 49, 60, 76, 95, 111, 160] [6, 61, 77, 78, 99, 151, 161] [7, 41, 62, 78, 97, 162, 182] [42, 63, 75, 79, 80, 153, 163] [26, 43, 64, 76, 80, 164, 184] [5, 44, 65, 77, 81, 155, 165] [5, 28, 45, 66, 78, 166, 186] [6, 7, 46, 67, 79, 157, 167] [9, 58, 68, 82, 83, 116, 168] [0, 10, 69, 83, 98, 117, 169] [11, 50, 70, 84, 85, 170, 187] [12, 33, 51, 71, 85, 171, 188] [8, 52, 72, 82, 86, 172, 189] [8, 35, 53, 73, 83, 173, 190] [9, 10, 54, 74, 84, 174, 191] [14, 67, 75, 87, 88, 104, 175] [1, 15, 76, 88, 99, 123, 176] [16, 59, 77, 89, 90, 169, 177] [17, 42, 60, 78, 90, 178, 192] [13, 61, 79, 87, 91, 171, 179] [13, 44, 62, 80, 88, 180, 194] [14, 15, 63, 81, 89, 173, 181] [3, 21, 82, 92, 93, 130, 182] [4, 22, 50, 83, 93, 131, 183] [18, 23, 68, 84, 94, 184, 195] [18, 24, 52, 69, 85, 185, 196] [19, 20, 70, 86, 92, 186, 197] [6, 28, 87, 95, 96, 112, 187] [7, 29, 59, 88, 96, 139, 188] [25, 30, 75, 89, 97, 183, 189] [25, 31, 61, 76, 90, 190, 198] [26, 27, 77, 91, 95, 185, 191] [11, 32, 37, 92, 98, 148, 192] [12, 32, 38, 68, 93, 149, 193] [33, 34, 39, 82, 94, 194, 199] [16, 41, 46, 95, 99, 124, 195] [17, 41, 47, 75, 96, 158, 196] [42, 43, 48, 87, 97, 193, 197] [23, 50, 51, 56, 98, 167, 198] [30, 59, 60, 65, 99, 140, 199]
Code ID 200-8-18 · download JSON · raw on GitHub