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[[156,2,17]] d ≤
n
156
k
2
d
17
kd²/n
3.705
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[6, 13, 49, 57, 60, 69, 82, 89, 95, 96, 104, 111, 112, 113, 136, 137, 139]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[7, 9, 20, 24, 28, 29, 30, 58, 61, 70, 73, 99, 105, 114, 123, 126, 147]
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)×78 (2,2)×117 (3,2)×234 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 78 (1,4): 78 (2,2): 117 (2,4): 546 (2,6): 468 (3,2): 234 (3,4): 2418 (3,6): 6162 (3,8): 4446 (3,10): 312
trapping sets H_Z (1,2)×78 (2,2)×117 (3,2)×234 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 78 (1,4): 78 (2,2): 117 (2,4): 546 (2,6): 468 (3,2): 234 (3,4): 2418 (3,6): 6162 (3,8): 4446 (3,10): 312

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

[[156,2,17]] — two-block group-algebra code on SmallGroup(78,3)

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 = 3.71.

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_order78_k2.txt, line 22. This row is SmallGroup(78,3) with nonidentity GAP supports a=[8], b=[5, 17, 46]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 17, not an exact certificate. The published distance (d=17) 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(78,3), 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] and b=[5, 17, 46] 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 = 156 - rank(HX) - rank(HZ) = 2. 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 78 (max weight 6) · Z-checks 78 (max weight 6)
H_X (78 checks, sparse supports)
[0, 76, 78, 105, 147, 148] [1, 74, 79, 102, 144, 145] [2, 8, 79, 80, 111, 152] [3, 66, 81, 119, 141, 155] [4, 5, 78, 82, 108, 150] [5, 77, 83, 116, 138, 154] [6, 14, 82, 83, 84, 117] [0, 7, 85, 89, 125, 147] [8, 72, 86, 112, 152, 153] [9, 10, 80, 81, 87, 114] [10, 11, 86, 88, 122, 144] [11, 69, 89, 109, 150, 151] [12, 20, 87, 88, 90, 123] [2, 13, 79, 91, 95, 131] [3, 14, 83, 92, 118, 155] [15, 16, 84, 85, 93, 120] [16, 17, 78, 92, 94, 128] [1, 17, 81, 95, 115, 154] [18, 26, 93, 94, 96, 129] [6, 19, 82, 97, 101, 137] [7, 20, 88, 89, 98, 124] [21, 22, 90, 91, 99, 126] [22, 23, 80, 98, 100, 134] [4, 23, 85, 86, 101, 121] [24, 32, 99, 100, 102, 135] [12, 25, 87, 103, 107, 143] [13, 26, 94, 95, 104, 130] [27, 28, 96, 97, 105, 132] [28, 29, 84, 104, 106, 140] [9, 29, 91, 92, 107, 127] [30, 38, 105, 106, 108, 141] [18, 31, 93, 109, 113, 149] [19, 32, 100, 101, 110, 136] [33, 34, 102, 103, 111, 138] [34, 35, 90, 110, 112, 146] [15, 35, 97, 98, 113, 133] [36, 44, 111, 112, 114, 147] [24, 37, 99, 115, 119, 153] [25, 38, 106, 107, 116, 142] [39, 40, 108, 109, 117, 144] [40, 41, 96, 116, 118, 151] [21, 41, 103, 104, 119, 139] [42, 50, 79, 117, 118, 120] [30, 43, 105, 121, 125, 155] [31, 44, 112, 113, 122, 148] [45, 46, 78, 114, 115, 123] [46, 47, 102, 122, 124, 154] [27, 47, 109, 110, 125, 145] [48, 56, 82, 123, 124, 126] [36, 49, 89, 111, 127, 131] [37, 50, 118, 119, 128, 152] [51, 52, 80, 120, 121, 129] [52, 53, 86, 108, 128, 130] [33, 53, 115, 116, 131, 150] [54, 62, 87, 129, 130, 132] [42, 55, 95, 117, 133, 137] [43, 56, 83, 124, 125, 134] [57, 58, 84, 126, 127, 135] [58, 59, 92, 114, 134, 136] [39, 59, 81, 121, 122, 137] [60, 68, 93, 135, 136, 138] [48, 61, 101, 123, 139, 143] [49, 62, 88, 130, 131, 140] [63, 64, 90, 132, 133, 141] [64, 65, 98, 120, 140, 142] [45, 65, 85, 127, 128, 143] [66, 73, 99, 141, 142, 144] [54, 67, 107, 129, 145, 149] [55, 68, 94, 136, 137, 146] [69, 70, 96, 138, 139, 147] [70, 71, 104, 126, 146, 148] [51, 71, 91, 133, 134, 149] [60, 72, 113, 135, 150, 153] [61, 73, 100, 142, 143, 151] [74, 75, 110, 132, 151, 152] [57, 75, 97, 139, 140, 153] [67, 76, 106, 148, 149, 154] [63, 77, 103, 145, 146, 155]
H_Z (78 checks, sparse supports)
[0, 4, 16, 45, 78, 85] [1, 2, 13, 42, 79, 95] [2, 9, 22, 51, 80, 91] [3, 9, 17, 59, 81, 92] [4, 6, 19, 48, 82, 101] [5, 6, 14, 56, 82, 83] [6, 15, 28, 57, 84, 97] [7, 15, 23, 65, 85, 98] [8, 10, 23, 52, 80, 86] [9, 12, 25, 54, 87, 107] [10, 12, 20, 62, 87, 88] [7, 11, 20, 49, 88, 89] [12, 21, 34, 63, 90, 103] [13, 21, 29, 71, 91, 104] [14, 16, 29, 58, 84, 92] [15, 18, 31, 60, 93, 113] [16, 18, 26, 68, 93, 94] [13, 17, 26, 55, 94, 95] [18, 27, 40, 69, 96, 109] [19, 27, 35, 75, 97, 110] [20, 22, 35, 64, 90, 98] [21, 24, 37, 66, 99, 119] [22, 24, 32, 73, 99, 100] [19, 23, 32, 61, 100, 101] [1, 24, 33, 46, 102, 115] [25, 33, 41, 77, 103, 116] [26, 28, 41, 70, 96, 104] [0, 27, 30, 43, 105, 125] [28, 30, 38, 76, 105, 106] [25, 29, 38, 67, 106, 107] [4, 30, 39, 52, 108, 121] [11, 31, 39, 47, 109, 122] [32, 34, 47, 74, 102, 110] [2, 33, 36, 49, 111, 131] [8, 34, 36, 44, 111, 112] [31, 35, 44, 72, 112, 113] [9, 36, 45, 58, 114, 127] [17, 37, 45, 53, 115, 128] [5, 38, 40, 53, 108, 116] [6, 39, 42, 55, 117, 137] [14, 40, 42, 50, 117, 118] [3, 37, 41, 50, 118, 119] [15, 42, 51, 64, 120, 133] [23, 43, 51, 59, 121, 134] [10, 44, 46, 59, 114, 122] [12, 45, 48, 61, 123, 143] [20, 46, 48, 56, 123, 124] [7, 43, 47, 56, 124, 125] [21, 48, 57, 70, 126, 139] [29, 49, 57, 65, 127, 140] [16, 50, 52, 65, 120, 128] [18, 51, 54, 67, 129, 149] [26, 52, 54, 62, 129, 130] [13, 49, 53, 62, 130, 131] [27, 54, 63, 74, 132, 145] [35, 55, 63, 71, 133, 146] [22, 56, 58, 71, 126, 134] [24, 57, 60, 72, 135, 153] [32, 58, 60, 68, 135, 136] [19, 55, 59, 68, 136, 137] [5, 33, 60, 69, 138, 150] [41, 61, 69, 75, 139, 151] [28, 62, 64, 75, 132, 140] [3, 30, 63, 66, 141, 155] [38, 64, 66, 73, 141, 142] [25, 61, 65, 73, 142, 143] [1, 10, 39, 66, 81, 144] [1, 47, 67, 77, 145, 154] [34, 68, 70, 77, 138, 146] [0, 7, 36, 69, 89, 147] [0, 44, 70, 76, 147, 148] [31, 67, 71, 76, 148, 149] [4, 11, 53, 72, 86, 150] [11, 40, 73, 74, 144, 151] [2, 8, 50, 74, 79, 152] [8, 37, 72, 75, 152, 153] [5, 17, 46, 76, 78, 154] [3, 14, 43, 77, 83, 155]
Code ID 156-2-17 · download JSON · raw on GitHub