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[[168,44,2]] d =
n
168
k
44
d
2
kd²/n
1.048
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)
[146, 155]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[23, 43]
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 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×84 (2,0)×84 (3,3)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 84 (1,4): 84 (2,0): 84 (2,3): 72 (2,4): 84 (2,5): 864 (2,6): 336 (3,3): 100 (3,4): 1452 (3,5): 1512 (3,6): 4352 (3,7): 8460 (3,8): 1680 (3,9): 864
trapping sets H_Z (1,3)×84 (2,0)×84 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 84 (1,4): 84 (2,0): 84 (2,3): 72 (2,4): 84 (2,5): 864 (2,6): 336 (3,0): 28 (3,2): 108 (3,3): 100 (3,4): 936 (3,5): 1368 (3,6): 4560 (3,7): 8856 (3,8): 2352 (3,9): 720

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_wt7wtL3_order84_k44.txt, line 1. nonidentity supports a=[3, 9], b=[2, 11, 16].
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

[[168,44,2]] — 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 = 1.05.

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_order84_k44.txt, line 1. This row is SmallGroup(84,1) with nonidentity GAP supports a=[3, 9], b=[2, 11, 16]. 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 ≤ 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=[3, 9], b=[2, 11, 16] 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) = 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 84 (max weight 7) · Z-checks 84 (max weight 7)
H_X (84 checks, sparse supports)
[0, 2, 8, 84, 90, 132, 151] [1, 5, 13, 84, 85, 115, 156] [0, 2, 8, 86, 98, 112, 131] [3, 9, 18, 85, 87, 121, 160] [4, 10, 19, 88, 92, 108, 159] [1, 5, 13, 86, 89, 96, 137] [6, 14, 24, 87, 90, 126, 163] [7, 15, 25, 91, 92, 97, 143] [0, 2, 8, 92, 108, 122, 141] [3, 9, 18, 89, 93, 101, 142] [4, 10, 19, 84, 90, 94, 165] [11, 20, 30, 95, 97, 102, 149] [12, 21, 31, 96, 115, 148, 163] [1, 5, 13, 92, 97, 105, 147] [6, 14, 24, 93, 98, 106, 148] [7, 15, 25, 84, 85, 99, 153] [16, 26, 36, 100, 102, 108, 154] [17, 27, 37, 101, 121, 131, 151] [3, 9, 18, 97, 102, 111, 152] [4, 10, 19, 86, 98, 103, 167] [11, 20, 30, 85, 87, 104, 158] [12, 21, 31, 96, 105, 148, 157] [22, 32, 42, 106, 126, 137, 156] [23, 33, 43, 107, 136, 139, 144] [6, 14, 24, 102, 108, 116, 157] [7, 15, 25, 86, 89, 109, 161] [16, 26, 36, 87, 90, 110, 162] [17, 27, 37, 101, 111, 131, 141] [28, 38, 48, 112, 132, 142, 160] [29, 39, 49, 113, 119, 127, 145] [11, 20, 30, 89, 93, 114, 164] [12, 21, 31, 105, 115, 157, 163] [22, 32, 42, 106, 116, 137, 147] [23, 33, 43, 117, 119, 124, 146] [34, 44, 54, 118, 125, 133, 150] [35, 45, 55, 119, 120, 124, 161] [16, 26, 36, 93, 98, 120, 166] [17, 27, 37, 111, 121, 141, 151] [28, 38, 48, 112, 122, 142, 152] [29, 39, 49, 107, 123, 125, 129] [40, 50, 60, 124, 130, 138, 155] [41, 51, 61, 103, 107, 125, 164] [22, 32, 42, 116, 126, 147, 156] [23, 33, 43, 127, 129, 134, 155] [34, 44, 54, 113, 128, 130, 135] [35, 45, 55, 100, 129, 134, 143] [46, 56, 66, 109, 113, 130, 166] [47, 57, 67, 103, 112, 131, 167] [28, 38, 48, 122, 132, 152, 160] [29, 39, 49, 117, 133, 135, 139] [40, 50, 60, 118, 134, 136, 140] [41, 51, 61, 88, 117, 135, 149] [52, 62, 71, 114, 118, 136, 167] [53, 63, 72, 96, 109, 137, 161] [34, 44, 54, 123, 138, 140, 145] [35, 45, 55, 110, 139, 144, 153] [46, 56, 66, 91, 123, 140, 154] [47, 57, 67, 88, 122, 141, 159] [58, 68, 76, 101, 114, 142, 164] [59, 69, 77, 100, 127, 143, 155] [40, 50, 60, 128, 144, 146, 150] [41, 51, 61, 94, 127, 145, 158] [52, 62, 71, 95, 128, 146, 159] [53, 63, 72, 91, 105, 143, 147] [64, 73, 79, 106, 120, 148, 166] [65, 74, 80, 88, 133, 139, 149] [46, 56, 66, 99, 133, 150, 162] [47, 57, 67, 94, 132, 151, 165] [58, 68, 76, 95, 111, 149, 152] [59, 69, 77, 107, 110, 136, 153] [70, 78, 82, 91, 138, 145, 154] [52, 62, 71, 104, 138, 155, 165] [53, 63, 72, 99, 115, 153, 156] [64, 73, 79, 100, 116, 154, 157] [65, 74, 80, 94, 113, 119, 158] [75, 81, 83, 95, 144, 150, 159] [58, 68, 76, 104, 121, 158, 160] [59, 69, 77, 117, 120, 146, 161] [70, 78, 82, 99, 118, 125, 162] [64, 73, 79, 110, 126, 162, 163] [65, 74, 80, 103, 123, 129, 164] [75, 81, 83, 104, 124, 130, 165] [70, 78, 82, 109, 128, 135, 166] [75, 81, 83, 114, 134, 140, 167]
H_Z (84 checks, sparse supports)
[0, 1, 10, 15, 84, 86, 92] [1, 3, 15, 20, 85, 89, 97] [2, 5, 19, 25, 84, 86, 92] [3, 6, 20, 26, 87, 93, 102] [4, 51, 57, 65, 88, 94, 103] [5, 9, 25, 30, 85, 89, 97] [0, 6, 10, 26, 90, 98, 108] [7, 56, 63, 70, 91, 99, 109] [4, 7, 8, 13, 84, 86, 92] [9, 14, 30, 36, 87, 93, 102] [10, 61, 67, 74, 88, 94, 103] [11, 62, 68, 75, 95, 104, 114] [5, 12, 21, 53, 96, 105, 115] [7, 11, 13, 18, 85, 89, 97] [2, 14, 19, 36, 90, 98, 108] [15, 66, 72, 78, 91, 99, 109] [16, 45, 59, 73, 100, 110, 120] [9, 17, 27, 58, 101, 111, 121] [11, 16, 18, 24, 87, 93, 102] [19, 41, 47, 80, 88, 94, 103] [20, 71, 76, 81, 95, 104, 114] [13, 21, 31, 63, 96, 105, 115] [14, 22, 32, 64, 106, 116, 126] [23, 39, 41, 69, 107, 117, 127] [4, 8, 16, 24, 90, 98, 108] [25, 46, 53, 82, 91, 99, 109] [26, 55, 69, 79, 100, 110, 120] [18, 27, 37, 68, 101, 111, 121] [2, 28, 38, 47, 112, 122, 132] [29, 44, 46, 74, 113, 123, 133] [30, 52, 58, 83, 95, 104, 114] [1, 12, 31, 72, 96, 105, 115] [24, 32, 42, 73, 106, 116, 126] [33, 49, 51, 77, 107, 117, 127] [34, 50, 52, 78, 118, 128, 138] [29, 33, 35, 74, 119, 129, 139] [35, 36, 64, 77, 100, 110, 120] [3, 17, 37, 76, 101, 111, 121] [8, 38, 48, 57, 112, 122, 132] [39, 54, 56, 80, 113, 123, 133] [33, 35, 40, 81, 124, 134, 144] [34, 39, 41, 78, 125, 135, 145] [6, 22, 42, 79, 106, 116, 126] [29, 43, 59, 61, 107, 117, 127] [44, 60, 62, 82, 118, 128, 138] [39, 43, 45, 80, 119, 129, 139] [40, 44, 46, 81, 130, 140, 150] [2, 17, 27, 47, 131, 141, 151] [0, 28, 48, 67, 112, 122, 132] [34, 49, 65, 66, 113, 123, 133] [43, 45, 50, 83, 124, 134, 144] [44, 49, 51, 82, 125, 135, 145] [23, 50, 52, 69, 136, 146, 155] [5, 22, 32, 53, 137, 147, 156] [40, 54, 70, 71, 118, 128, 138] [23, 49, 55, 65, 119, 129, 139] [50, 54, 56, 83, 130, 140, 150] [8, 27, 37, 57, 131, 141, 151] [9, 28, 38, 58, 142, 152, 160] [7, 45, 59, 63, 143, 153, 161] [23, 55, 60, 75, 124, 134, 144] [29, 54, 61, 70, 125, 135, 145] [33, 60, 62, 77, 136, 146, 155] [13, 32, 42, 63, 137, 147, 156] [12, 14, 21, 64, 148, 157, 163] [11, 51, 65, 68, 149, 158, 164] [34, 60, 66, 75, 130, 140, 150] [0, 17, 37, 67, 131, 141, 151] [18, 38, 48, 68, 142, 152, 160] [15, 55, 69, 72, 143, 153, 161] [16, 56, 70, 73, 154, 162, 166] [40, 43, 59, 71, 136, 146, 155] [1, 22, 42, 72, 137, 147, 156] [21, 24, 31, 73, 148, 157, 163] [20, 61, 74, 76, 149, 158, 164] [4, 57, 62, 75, 159, 165, 167] [3, 28, 48, 76, 142, 152, 160] [25, 35, 53, 77, 143, 153, 161] [26, 66, 78, 79, 154, 162, 166] [6, 12, 31, 79, 148, 157, 163] [30, 41, 58, 80, 149, 158, 164] [10, 67, 71, 81, 159, 165, 167] [36, 46, 64, 82, 154, 162, 166] [19, 47, 52, 83, 159, 165, 167]
Code ID 168-44-2 · download JSON · raw on GitHub