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[[168,12,4]] d =
n
168
k
12
d
4
kd²/n
1.143
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 5, w_Z = 5 (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)
[96, 101, 106, 112]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[45, 51, 56, 62]
certificate exact, d = 4 · scipy/HiGHS cutoff IP
X: no logical < 4 exists; Z: no logical < 4 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×84 (2,2)×84 (3,2)×84 (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 (1,3): 84 (2,2): 84 (2,3): 504 (2,4): 252 (3,2): 84 (3,3): 1568 (3,4): 2772 (3,5): 1092 (3,6): 504 (3,7): 84
trapping sets H_Z (1,2)×84 (2,2)×84 (3,2)×84 (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 (1,3): 84 (2,2): 84 (2,3): 504 (2,4): 252 (3,2): 84 (3,3): 1568 (3,4): 2772 (3,5): 1092 (3,6): 504 (3,7): 84

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_wt5wtL2_order84_k12.txt, line 1. nonidentity supports a=[2], b=[3, 5].
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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[168,12,4]] — two-block group-algebra code on SmallGroup(84,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.14.

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_wt5wtL2_order84_k12.txt, line 1. This row is SmallGroup(84,1) with nonidentity GAP supports a=[2], b=[3, 5]. 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=12), 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

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