[[168,24,6]] · QEC Challenge

[[168,24,6]] d ≤

168

kd²/n

5.143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[7, 85, 92, 103, 115, 149]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[21, 57, 61, 69, 86, 117]

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 6 · H_Z 6 (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)×84 (2,2)×84 (3,2)×168 (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 (1,4): 84 (2,2): 84 (2,4): 672 (2,6): 504 (3,2): 168 (3,4): 2856 (3,6): 7056 (3,8): 3780 (3,10): 336

trapping sets H_Z (1,2)×84 (2,2)×84 (3,2)×168 (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 (1,4): 84 (2,2): 84 (2,4): 672 (2,6): 504 (3,2): 168 (3,4): 2856 (3,6): 7056 (3,8): 3780 (3,10): 336

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

[[168,24,6]] — two-block group-algebra code on SmallGroup(84,4)

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 = 5.14.

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_wt6wtL2_order84_k24.txt, line 10. This row is SmallGroup(84,4) with nonidentity GAP supports a=[5], b=[6, 13, 40]. 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=24), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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(84,4), 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=[5] and b=[6, 13, 40] 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) = 24. 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 6) · Z-checks 84 (max weight 6)

H_X (84 checks, sparse supports)

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

H_Z (84 checks, sparse supports)

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

Code ID 168-24-6 · download JSON · raw on GitHub