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[[168,56,2]] d =
n
168
k
56
d
2
kd²/n
1.333
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[117, 128]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[23, 34]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 exists

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 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)×84 (2,0)×42 (3,2)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 84 (1,5): 84 (2,0): 42 (2,3): 168 (2,4): 84 (2,5): 504 (2,6): 126 (2,8): 336 (3,2): 336 (3,3): 1008 (3,4): 252 (3,5): 1512 (3,6): 2184 (3,7): 1260 (3,8): 4620 (3,9): 1204 (3,11): 2016
trapping sets H_Z (1,2)×84 (2,0)×42 (3,2)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 84 (1,5): 84 (2,0): 42 (2,3): 168 (2,4): 84 (2,5): 504 (2,6): 126 (2,8): 336 (3,2): 336 (3,3): 1008 (3,4): 252 (3,5): 1512 (3,6): 2184 (3,7): 1260 (3,8): 4620 (3,9): 1204 (3,11): 2016

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(84,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL2_order84_k56.txt, line 1. nonidentity supports a=[4], b=[2, 4, 16, 73].
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
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

[[168,56,2]] — two-block group-algebra code on SmallGroup(84,1)

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 = 1.33.

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_order84_k56.txt, line 1. This row is SmallGroup(84,1) with nonidentity GAP supports a=[4], b=[2, 4, 16, 73]. 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=56), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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

DeepSeek V4 Flash 0731 (provenance.model), 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(84,1), 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=[4], b=[2, 4, 16, 73] 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 = 168 - rank(HX) - rank(HZ) = 56. 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 84 (max weight 7) · Z-checks 84 (max weight 7)
H_X (84 checks, sparse supports)
[0, 3, 84, 87, 90, 132, 165] [1, 6, 84, 85, 90, 115, 153] [2, 9, 86, 93, 98, 112, 167] [0, 3, 84, 85, 87, 121, 158] [4, 11, 88, 95, 108, 122, 159] [5, 14, 86, 89, 96, 98, 161] [1, 6, 85, 87, 90, 126, 162] [7, 16, 91, 92, 100, 105, 143] [8, 18, 92, 102, 108, 122, 159] [2, 9, 86, 89, 93, 101, 164] [10, 20, 90, 94, 104, 132, 165] [4, 11, 88, 95, 97, 111, 149] [12, 22, 96, 106, 148, 157, 163] [13, 24, 92, 97, 105, 108, 143] [5, 14, 89, 93, 98, 106, 166] [15, 26, 84, 99, 110, 115, 153] [7, 16, 91, 100, 102, 116, 154] [17, 28, 101, 112, 131, 141, 151] [8, 18, 92, 97, 102, 111, 149] [19, 30, 98, 103, 112, 114, 167] [10, 20, 85, 94, 104, 121, 158] [21, 32, 105, 116, 148, 157, 163] [12, 22, 96, 106, 137, 147, 156] [23, 34, 107, 110, 118, 136, 144] [13, 24, 97, 102, 108, 116, 154] [25, 36, 86, 96, 109, 120, 161] [15, 26, 87, 99, 110, 126, 162] [27, 38, 111, 122, 131, 141, 151] [17, 28, 101, 112, 142, 152, 160] [29, 40, 94, 113, 119, 124, 127] [19, 30, 89, 101, 103, 114, 164] [31, 42, 115, 126, 148, 157, 163] [21, 32, 105, 116, 137, 147, 156] [33, 44, 117, 120, 124, 128, 146] [23, 34, 99, 107, 118, 125, 133] [35, 46, 119, 120, 124, 130, 146] [25, 36, 93, 106, 109, 120, 166] [37, 48, 121, 131, 132, 141, 151] [27, 38, 111, 122, 142, 152, 160] [39, 50, 103, 107, 123, 129, 134] [29, 40, 104, 113, 124, 130, 138] [41, 52, 103, 107, 125, 129, 136] [31, 42, 115, 126, 137, 147, 156] [43, 54, 100, 127, 134, 138, 155] [33, 44, 109, 113, 117, 128, 135] [45, 56, 100, 129, 134, 140, 155] [35, 46, 109, 113, 119, 130, 135] [47, 58, 98, 112, 131, 142, 167] [37, 48, 121, 132, 142, 152, 160] [49, 60, 88, 117, 133, 139, 144] [39, 50, 114, 118, 123, 134, 140] [51, 62, 88, 117, 135, 139, 146] [41, 52, 114, 118, 125, 136, 140] [53, 64, 86, 96, 137, 148, 161] [43, 54, 91, 123, 127, 138, 145] [55, 66, 110, 136, 139, 144, 150] [45, 56, 91, 123, 129, 140, 145] [57, 68, 108, 122, 141, 152, 159] [47, 58, 89, 101, 131, 142, 164] [59, 70, 100, 134, 143, 154, 155] [49, 60, 95, 128, 133, 144, 150] [61, 71, 94, 119, 127, 145, 155] [51, 62, 95, 128, 135, 146, 150] [63, 73, 92, 105, 143, 147, 157] [53, 64, 93, 106, 137, 148, 166] [65, 75, 88, 117, 139, 149, 159] [55, 66, 99, 125, 133, 139, 150] [67, 76, 90, 132, 151, 160, 165] [57, 68, 97, 111, 141, 149, 152] [69, 78, 110, 136, 144, 153, 162] [59, 70, 91, 123, 143, 145, 154] [61, 71, 104, 130, 138, 145, 155] [72, 79, 84, 115, 153, 156, 163] [63, 73, 102, 116, 147, 154, 157] [74, 81, 94, 119, 127, 158, 165] [65, 75, 95, 128, 149, 150, 159] [67, 76, 85, 121, 151, 158, 160] [77, 82, 120, 124, 146, 161, 166] [69, 78, 99, 125, 133, 153, 162] [72, 79, 87, 126, 156, 162, 163] [80, 83, 103, 107, 129, 164, 167] [74, 81, 104, 130, 138, 158, 165] [77, 82, 109, 113, 135, 161, 166] [80, 83, 114, 118, 140, 164, 167]
H_Z (84 checks, sparse supports)
[0, 1, 3, 15, 72, 84, 87] [1, 3, 6, 20, 76, 85, 90] [2, 5, 9, 25, 53, 86, 93] [0, 3, 6, 26, 79, 84, 87] [4, 11, 49, 51, 65, 88, 95] [5, 9, 14, 30, 58, 89, 98] [0, 1, 6, 10, 67, 85, 90] [7, 16, 54, 56, 70, 91, 100] [7, 8, 13, 18, 63, 92, 102] [2, 9, 14, 36, 64, 86, 93] [10, 20, 29, 61, 74, 94, 104] [4, 11, 60, 62, 75, 88, 95] [5, 12, 22, 25, 53, 96, 106] [11, 13, 18, 24, 68, 97, 108] [2, 5, 14, 19, 47, 89, 98] [15, 26, 34, 66, 78, 99, 110] [7, 16, 43, 45, 59, 91, 100] [9, 17, 28, 30, 58, 101, 112] [8, 16, 18, 24, 73, 92, 102] [19, 30, 39, 41, 80, 103, 114] [10, 20, 40, 71, 81, 94, 104] [7, 13, 21, 32, 63, 105, 116] [12, 14, 22, 36, 64, 96, 106] [23, 34, 39, 41, 80, 107, 118] [4, 8, 13, 24, 57, 97, 108] [25, 36, 44, 46, 82, 109, 120] [15, 23, 26, 55, 69, 99, 110] [11, 18, 27, 38, 68, 111, 122] [2, 17, 19, 28, 47, 101, 112] [29, 40, 44, 46, 82, 113, 124] [19, 30, 50, 52, 83, 103, 114] [1, 15, 31, 42, 72, 115, 126] [16, 21, 24, 32, 73, 105, 116] [33, 44, 49, 51, 65, 117, 128] [23, 34, 50, 52, 83, 107, 118] [29, 35, 46, 61, 74, 119, 130] [25, 33, 35, 36, 77, 109, 120] [3, 20, 37, 48, 76, 121, 132] [4, 8, 27, 38, 57, 111, 122] [39, 50, 54, 56, 70, 123, 134] [29, 33, 35, 40, 77, 113, 124] [34, 41, 52, 66, 78, 125, 136] [6, 26, 31, 42, 79, 115, 126] [29, 43, 54, 61, 74, 127, 138] [33, 44, 60, 62, 75, 117, 128] [39, 41, 45, 56, 80, 129, 140] [35, 40, 46, 71, 81, 119, 130] [17, 27, 37, 47, 58, 131, 142] [0, 10, 37, 48, 67, 121, 132] [34, 49, 60, 66, 78, 133, 144] [39, 43, 45, 50, 59, 123, 134] [44, 46, 51, 62, 82, 135, 146] [23, 41, 52, 55, 69, 125, 136] [22, 32, 42, 53, 64, 137, 148] [40, 43, 54, 71, 81, 127, 138] [49, 51, 55, 65, 66, 139, 150] [45, 50, 52, 56, 83, 129, 140] [17, 27, 37, 57, 68, 141, 152] [28, 38, 47, 48, 58, 131, 142] [7, 13, 59, 63, 70, 143, 154] [23, 49, 55, 60, 69, 133, 144] [54, 56, 61, 70, 71, 145, 155] [33, 35, 51, 62, 77, 135, 146] [22, 32, 42, 63, 73, 147, 157] [12, 21, 31, 53, 64, 137, 148] [11, 18, 65, 68, 75, 149, 159] [55, 60, 62, 66, 75, 139, 150] [17, 27, 37, 67, 76, 151, 160] [28, 38, 48, 57, 68, 141, 152] [1, 15, 69, 72, 78, 153, 162] [16, 24, 59, 70, 73, 143, 154] [43, 45, 59, 61, 71, 145, 155] [22, 32, 42, 72, 79, 156, 163] [12, 21, 31, 63, 73, 147, 157] [3, 20, 74, 76, 81, 158, 165] [4, 8, 57, 65, 75, 149, 159] [28, 38, 48, 67, 76, 151, 160] [5, 25, 53, 77, 82, 161, 166] [6, 26, 69, 78, 79, 153, 162] [12, 21, 31, 72, 79, 156, 163] [9, 30, 58, 80, 83, 164, 167] [0, 10, 67, 74, 81, 158, 165] [14, 36, 64, 77, 82, 161, 166] [2, 19, 47, 80, 83, 164, 167]
Code ID 168-56-2 · download JSON · raw on GitHub