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[[136,68,2]] d =
n
136
k
68
d
2
kd²/n
2.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 8, w_Z = 8 (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)
[1, 4]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[133, 135]
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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×68 (2,0)×68 (3,2)×1564 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 68 (1,6): 68 (2,0): 68 (2,4): 544 (2,8): 136 (3,2): 1564 (3,6): 2924 (3,10): 544
trapping sets H_Z (1,2)×68 (2,0)×68 (3,2)×1564 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 68 (1,6): 68 (2,0): 68 (2,4): 544 (2,8): 136 (3,2): 1564 (3,6): 2924 (3,10): 544

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(68,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order68_k68.txt, line 1. nonidentity supports a=[3], b=[2, 3, 4, 5, 7].
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

[[136,68,2]] — two-block group-algebra code on SmallGroup(68,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.00.

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_wt8wtL2_order68_k68.txt, line 1. This row is SmallGroup(68,1) with nonidentity GAP supports a=[3], b=[2, 3, 4, 5, 7]. 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=68), 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 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(68,1), 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], b=[2, 3, 4, 5, 7] 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 = 136 - rank(HX) - rank(HZ) = 68. 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 68 (max weight 8) · Z-checks 68 (max weight 8)
H_X (68 checks, sparse supports)
[0, 2, 68, 69, 70, 72, 131, 134] [1, 4, 68, 69, 70, 72, 133, 135] [0, 2, 68, 69, 70, 72, 131, 134] [3, 6, 68, 70, 71, 74, 133, 135] [1, 4, 68, 69, 70, 72, 133, 135] [5, 8, 69, 72, 73, 76, 131, 134] [3, 6, 68, 70, 71, 74, 133, 135] [7, 10, 71, 74, 75, 78, 129, 132] [5, 8, 69, 72, 73, 76, 131, 134] [9, 12, 73, 76, 77, 80, 127, 130] [7, 10, 71, 74, 75, 78, 129, 132] [11, 14, 75, 78, 79, 82, 125, 128] [9, 12, 73, 76, 77, 80, 127, 130] [13, 16, 77, 80, 81, 84, 123, 126] [11, 14, 75, 78, 79, 82, 125, 128] [15, 18, 79, 82, 83, 86, 121, 124] [13, 16, 77, 80, 81, 84, 123, 126] [17, 20, 81, 84, 85, 88, 119, 122] [15, 18, 79, 82, 83, 86, 121, 124] [19, 22, 83, 86, 87, 90, 117, 120] [17, 20, 81, 84, 85, 88, 119, 122] [21, 24, 85, 88, 89, 92, 115, 118] [19, 22, 83, 86, 87, 90, 117, 120] [23, 26, 87, 90, 91, 94, 113, 116] [21, 24, 85, 88, 89, 92, 115, 118] [25, 28, 89, 92, 93, 96, 111, 114] [23, 26, 87, 90, 91, 94, 113, 116] [27, 30, 91, 94, 95, 98, 109, 112] [25, 28, 89, 92, 93, 96, 111, 114] [29, 32, 93, 96, 97, 100, 107, 110] [27, 30, 91, 94, 95, 98, 109, 112] [31, 34, 95, 98, 99, 102, 105, 108] [29, 32, 93, 96, 97, 100, 107, 110] [33, 36, 97, 100, 101, 103, 104, 106] [31, 34, 95, 98, 99, 102, 105, 108] [35, 38, 99, 101, 102, 103, 104, 106] [33, 36, 97, 100, 101, 103, 104, 106] [37, 40, 99, 101, 102, 104, 105, 108] [35, 38, 99, 101, 102, 103, 104, 106] [39, 42, 97, 100, 103, 106, 107, 110] [37, 40, 99, 101, 102, 104, 105, 108] [41, 44, 95, 98, 105, 108, 109, 112] [39, 42, 97, 100, 103, 106, 107, 110] [43, 46, 93, 96, 107, 110, 111, 114] [41, 44, 95, 98, 105, 108, 109, 112] [45, 48, 91, 94, 109, 112, 113, 116] [43, 46, 93, 96, 107, 110, 111, 114] [47, 50, 89, 92, 111, 114, 115, 118] [45, 48, 91, 94, 109, 112, 113, 116] [49, 52, 87, 90, 113, 116, 117, 120] [47, 50, 89, 92, 111, 114, 115, 118] [51, 54, 85, 88, 115, 118, 119, 122] [49, 52, 87, 90, 113, 116, 117, 120] [53, 56, 83, 86, 117, 120, 121, 124] [51, 54, 85, 88, 115, 118, 119, 122] [55, 58, 81, 84, 119, 122, 123, 126] [53, 56, 83, 86, 117, 120, 121, 124] [57, 60, 79, 82, 121, 124, 125, 128] [55, 58, 81, 84, 119, 122, 123, 126] [59, 62, 77, 80, 123, 126, 127, 130] [57, 60, 79, 82, 121, 124, 125, 128] [61, 64, 75, 78, 125, 128, 129, 132] [59, 62, 77, 80, 123, 126, 127, 130] [63, 66, 73, 76, 127, 130, 131, 134] [61, 64, 75, 78, 125, 128, 129, 132] [65, 67, 71, 74, 129, 132, 133, 135] [63, 66, 73, 76, 127, 130, 131, 134] [65, 67, 71, 74, 129, 132, 133, 135]
H_Z (68 checks, sparse supports)
[0, 1, 2, 3, 4, 6, 68, 70] [0, 1, 2, 4, 5, 8, 69, 72] [0, 1, 2, 3, 4, 6, 68, 70] [3, 6, 7, 10, 65, 67, 71, 74] [0, 1, 2, 4, 5, 8, 69, 72] [5, 8, 9, 12, 63, 66, 73, 76] [3, 6, 7, 10, 65, 67, 71, 74] [7, 10, 11, 14, 61, 64, 75, 78] [5, 8, 9, 12, 63, 66, 73, 76] [9, 12, 13, 16, 59, 62, 77, 80] [7, 10, 11, 14, 61, 64, 75, 78] [11, 14, 15, 18, 57, 60, 79, 82] [9, 12, 13, 16, 59, 62, 77, 80] [13, 16, 17, 20, 55, 58, 81, 84] [11, 14, 15, 18, 57, 60, 79, 82] [15, 18, 19, 22, 53, 56, 83, 86] [13, 16, 17, 20, 55, 58, 81, 84] [17, 20, 21, 24, 51, 54, 85, 88] [15, 18, 19, 22, 53, 56, 83, 86] [19, 22, 23, 26, 49, 52, 87, 90] [17, 20, 21, 24, 51, 54, 85, 88] [21, 24, 25, 28, 47, 50, 89, 92] [19, 22, 23, 26, 49, 52, 87, 90] [23, 26, 27, 30, 45, 48, 91, 94] [21, 24, 25, 28, 47, 50, 89, 92] [25, 28, 29, 32, 43, 46, 93, 96] [23, 26, 27, 30, 45, 48, 91, 94] [27, 30, 31, 34, 41, 44, 95, 98] [25, 28, 29, 32, 43, 46, 93, 96] [29, 32, 33, 36, 39, 42, 97, 100] [27, 30, 31, 34, 41, 44, 95, 98] [31, 34, 35, 37, 38, 40, 99, 102] [29, 32, 33, 36, 39, 42, 97, 100] [33, 35, 36, 37, 38, 40, 101, 104] [31, 34, 35, 37, 38, 40, 99, 102] [33, 35, 36, 38, 39, 42, 103, 106] [33, 35, 36, 37, 38, 40, 101, 104] [31, 34, 37, 40, 41, 44, 105, 108] [33, 35, 36, 38, 39, 42, 103, 106] [29, 32, 39, 42, 43, 46, 107, 110] [31, 34, 37, 40, 41, 44, 105, 108] [27, 30, 41, 44, 45, 48, 109, 112] [29, 32, 39, 42, 43, 46, 107, 110] [25, 28, 43, 46, 47, 50, 111, 114] [27, 30, 41, 44, 45, 48, 109, 112] [23, 26, 45, 48, 49, 52, 113, 116] [25, 28, 43, 46, 47, 50, 111, 114] [21, 24, 47, 50, 51, 54, 115, 118] [23, 26, 45, 48, 49, 52, 113, 116] [19, 22, 49, 52, 53, 56, 117, 120] [21, 24, 47, 50, 51, 54, 115, 118] [17, 20, 51, 54, 55, 58, 119, 122] [19, 22, 49, 52, 53, 56, 117, 120] [15, 18, 53, 56, 57, 60, 121, 124] [17, 20, 51, 54, 55, 58, 119, 122] [13, 16, 55, 58, 59, 62, 123, 126] [15, 18, 53, 56, 57, 60, 121, 124] [11, 14, 57, 60, 61, 64, 125, 128] [13, 16, 55, 58, 59, 62, 123, 126] [9, 12, 59, 62, 63, 66, 127, 130] [11, 14, 57, 60, 61, 64, 125, 128] [7, 10, 61, 64, 65, 67, 129, 132] [9, 12, 59, 62, 63, 66, 127, 130] [0, 2, 5, 8, 63, 66, 131, 134] [7, 10, 61, 64, 65, 67, 129, 132] [1, 3, 4, 6, 65, 67, 133, 135] [0, 2, 5, 8, 63, 66, 131, 134] [1, 3, 4, 6, 65, 67, 133, 135]
Code ID 136-68-2 · download JSON · raw on GitHub