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[[176,44,4]] d ≤
n
176
k
44
d
4
kd²/n
4.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[68, 77, 144, 153]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[68, 77, 144, 153]
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 4 · H_Z 4 (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 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×88 (2,2)×176 (3,2)×792 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 88 (1,4): 88 (2,2): 176 (2,4): 572 (2,6): 440 (3,2): 792 (3,4): 2640 (3,6): 4048 (3,8): 1936 (3,10): 176
trapping sets H_Z (1,2)×88 (2,2)×176 (3,2)×792 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 88 (1,4): 88 (2,2): 176 (2,4): 572 (2,6): 440 (3,2): 792 (3,4): 2640 (3,6): 4048 (3,8): 1936 (3,10): 176

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

[[176,44,4]] — two-block group-algebra code on SmallGroup(88,9)

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 = 4.00.

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_wt6wtL2_order88_k44.txt, line 8. This row is SmallGroup(88,9) with nonidentity GAP supports a=[6], b=[5, 7, 9]. 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=44), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance (d=4) 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(88,9), 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, 7, 9] 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 = 176 - rank(HX) - rank(HZ) = 44. 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 88 (max weight 6) · Z-checks 88 (max weight 6)
H_X (88 checks, sparse supports)
[0, 13, 88, 92, 166, 168] [1, 2, 89, 95, 162, 171] [2, 7, 90, 97, 162, 175] [3, 21, 89, 90, 91, 99] [4, 5, 88, 92, 173, 174] [0, 5, 93, 101, 166, 174] [6, 8, 88, 93, 94, 103] [7, 9, 89, 95, 170, 175] [8, 15, 88, 96, 101, 105] [1, 9, 90, 97, 170, 171] [10, 29, 94, 96, 98, 107] [11, 12, 91, 95, 97, 99] [3, 12, 89, 97, 100, 109] [4, 13, 93, 101, 168, 173] [14, 16, 91, 100, 102, 111] [15, 17, 92, 94, 101, 103] [16, 23, 91, 104, 109, 113] [6, 17, 92, 93, 96, 105] [18, 37, 102, 104, 106, 115] [19, 20, 98, 103, 105, 107] [10, 20, 94, 105, 108, 117] [11, 21, 90, 95, 100, 109] [22, 24, 98, 108, 110, 119] [23, 25, 99, 102, 109, 111] [24, 31, 98, 112, 117, 121] [14, 25, 99, 100, 104, 113] [26, 45, 110, 112, 114, 123] [27, 28, 106, 111, 113, 115] [18, 28, 102, 113, 116, 125] [19, 29, 96, 103, 108, 117] [30, 32, 106, 116, 118, 127] [31, 33, 107, 110, 117, 119] [32, 39, 106, 120, 125, 129] [22, 33, 107, 108, 112, 121] [34, 53, 118, 120, 122, 131] [35, 36, 114, 119, 121, 123] [26, 36, 110, 121, 124, 133] [27, 37, 104, 111, 116, 125] [38, 40, 114, 124, 126, 135] [39, 41, 115, 118, 125, 127] [40, 47, 114, 128, 133, 137] [30, 41, 115, 116, 120, 129] [42, 61, 126, 128, 130, 139] [43, 44, 122, 127, 129, 131] [34, 44, 118, 129, 132, 141] [35, 45, 112, 119, 124, 133] [46, 48, 122, 132, 134, 143] [47, 49, 123, 126, 133, 135] [48, 55, 122, 136, 141, 145] [38, 49, 123, 124, 128, 137] [50, 69, 134, 136, 138, 147] [51, 52, 130, 135, 137, 139] [42, 52, 126, 137, 140, 149] [43, 53, 120, 127, 132, 141] [54, 56, 130, 140, 142, 151] [55, 57, 131, 134, 141, 143] [56, 63, 130, 144, 149, 153] [46, 57, 131, 132, 136, 145] [58, 77, 142, 144, 146, 155] [59, 60, 138, 143, 145, 147] [50, 60, 134, 145, 148, 157] [51, 61, 128, 135, 140, 149] [62, 64, 138, 148, 150, 159] [63, 65, 139, 142, 149, 151] [64, 71, 138, 152, 157, 161] [54, 65, 139, 140, 144, 153] [66, 84, 150, 152, 154, 163] [67, 68, 146, 151, 153, 155] [58, 68, 142, 153, 156, 165] [59, 69, 136, 143, 148, 157] [70, 72, 146, 156, 158, 167] [71, 73, 147, 150, 157, 159] [72, 79, 146, 160, 165, 169] [62, 73, 147, 148, 152, 161] [74, 87, 158, 160, 162, 170] [75, 76, 154, 159, 161, 163] [66, 76, 150, 161, 164, 172] [67, 77, 144, 151, 156, 165] [78, 80, 154, 164, 166, 173] [79, 81, 155, 158, 165, 167] [80, 85, 154, 168, 172, 174] [70, 81, 155, 156, 160, 169] [82, 83, 162, 167, 169, 170] [74, 83, 158, 169, 171, 175] [75, 84, 152, 159, 164, 172] [85, 86, 163, 166, 172, 173] [78, 86, 163, 164, 168, 174] [82, 87, 160, 167, 171, 175]
H_Z (88 checks, sparse supports)
[0, 4, 6, 8, 88, 93] [1, 3, 7, 12, 89, 97] [2, 3, 9, 21, 89, 90] [3, 11, 14, 16, 91, 100] [0, 4, 15, 17, 92, 101] [5, 6, 13, 17, 92, 93] [6, 10, 15, 20, 94, 105] [1, 7, 11, 21, 90, 95] [8, 10, 17, 29, 94, 96] [2, 9, 11, 12, 95, 97] [10, 19, 22, 24, 98, 108] [3, 11, 23, 25, 99, 109] [12, 14, 21, 25, 99, 100] [5, 8, 13, 15, 88, 101] [14, 18, 23, 28, 102, 113] [6, 15, 19, 29, 96, 103] [16, 18, 25, 37, 102, 104] [8, 17, 19, 20, 103, 105] [18, 27, 30, 32, 106, 116] [10, 19, 31, 33, 107, 117] [20, 22, 29, 33, 107, 108] [12, 16, 21, 23, 91, 109] [22, 26, 31, 36, 110, 121] [14, 23, 27, 37, 104, 111] [24, 26, 33, 45, 110, 112] [16, 25, 27, 28, 111, 113] [26, 35, 38, 40, 114, 124] [18, 27, 39, 41, 115, 125] [28, 30, 37, 41, 115, 116] [20, 24, 29, 31, 98, 117] [30, 34, 39, 44, 118, 129] [22, 31, 35, 45, 112, 119] [32, 34, 41, 53, 118, 120] [24, 33, 35, 36, 119, 121] [34, 43, 46, 48, 122, 132] [26, 35, 47, 49, 123, 133] [36, 38, 45, 49, 123, 124] [28, 32, 37, 39, 106, 125] [38, 42, 47, 52, 126, 137] [30, 39, 43, 53, 120, 127] [40, 42, 49, 61, 126, 128] [32, 41, 43, 44, 127, 129] [42, 51, 54, 56, 130, 140] [34, 43, 55, 57, 131, 141] [44, 46, 53, 57, 131, 132] [36, 40, 45, 47, 114, 133] [46, 50, 55, 60, 134, 145] [38, 47, 51, 61, 128, 135] [48, 50, 57, 69, 134, 136] [40, 49, 51, 52, 135, 137] [50, 59, 62, 64, 138, 148] [42, 51, 63, 65, 139, 149] [52, 54, 61, 65, 139, 140] [44, 48, 53, 55, 122, 141] [54, 58, 63, 68, 142, 153] [46, 55, 59, 69, 136, 143] [56, 58, 65, 77, 142, 144] [48, 57, 59, 60, 143, 145] [58, 67, 70, 72, 146, 156] [50, 59, 71, 73, 147, 157] [60, 62, 69, 73, 147, 148] [52, 56, 61, 63, 130, 149] [62, 66, 71, 76, 150, 161] [54, 63, 67, 77, 144, 151] [64, 66, 73, 84, 150, 152] [56, 65, 67, 68, 151, 153] [66, 75, 78, 80, 154, 164] [58, 67, 79, 81, 155, 165] [68, 70, 77, 81, 155, 156] [60, 64, 69, 71, 138, 157] [70, 74, 79, 83, 158, 169] [62, 71, 75, 84, 152, 159] [72, 74, 81, 87, 158, 160] [64, 73, 75, 76, 159, 161] [1, 2, 74, 82, 162, 171] [66, 75, 85, 86, 163, 172] [76, 78, 84, 86, 163, 164] [68, 72, 77, 79, 146, 165] [0, 5, 78, 85, 166, 174] [70, 79, 82, 87, 160, 167] [0, 13, 80, 86, 166, 168] [72, 81, 82, 83, 167, 169] [7, 9, 74, 82, 170, 175] [1, 9, 83, 87, 170, 171] [76, 80, 84, 85, 154, 172] [4, 13, 78, 85, 168, 173] [4, 5, 80, 86, 173, 174] [2, 7, 83, 87, 162, 175]
Code ID 176-44-4 · download JSON · raw on GitHub