[[168,18,10]] · QEC Challenge

[[168,18,10]] d ≤

168

kd²/n

10.714

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[85, 97, 101, 109, 113, 123, 133, 147, 158, 164]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[0, 10, 33, 35, 43, 45, 47, 55, 57, 77]

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 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 (2,4)×336 (3,3)×84 (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 (2,4): 336 (2,5): 1008 (2,6): 336 (3,3): 84 (3,4): 168 (3,5): 1764 (3,6): 9324 (3,7): 10668 (3,8): 2436 (3,9): 1008 (3,10): 84

trapping sets H_Z (1,3)×84 (2,4)×336 (3,3)×84 (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 (2,4): 336 (2,5): 1008 (2,6): 336 (3,3): 84 (3,4): 168 (3,5): 1764 (3,6): 9324 (3,7): 10668 (3,8): 2436 (3,9): 1008 (3,10): 84

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 ≤ 8 (computed)

How this code was found

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

[[168,18,10]] — two-block group-algebra code on SmallGroup(84,2)

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 = 10.71.

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_k18.txt, line 6. This row is SmallGroup(84,2) with nonidentity GAP supports a=[3, 20], b=[8, 23, 53]. 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=18), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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(84,2), 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, 20] and b=[8, 23, 53] 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) = 18. 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, 8, 57, 84, 113, 142, 159] [1, 13, 63, 85, 118, 143, 148] [0, 2, 43, 86, 123, 152, 165] [3, 18, 68, 87, 124, 131, 149] [4, 19, 69, 88, 90, 125, 154] [1, 5, 49, 89, 128, 153, 157] [6, 24, 73, 90, 107, 137, 154] [7, 25, 74, 84, 91, 130, 159] [2, 8, 59, 92, 133, 160, 167] [3, 9, 54, 93, 134, 141, 158] [4, 10, 55, 94, 98, 135, 162] [11, 30, 78, 85, 95, 136, 143] [2, 12, 31, 87, 96, 100, 137] [5, 13, 65, 97, 138, 161, 163] [6, 14, 60, 98, 117, 147, 162] [7, 15, 61, 86, 99, 140, 165] [16, 36, 81, 87, 100, 119, 149] [5, 17, 37, 88, 90, 101, 142] [9, 18, 70, 102, 144, 151, 164] [0, 10, 19, 103, 108, 145, 166] [11, 20, 66, 89, 104, 146, 153] [12, 21, 67, 93, 105, 110, 147] [9, 22, 42, 84, 91, 106, 148] [10, 23, 43, 95, 107, 112, 149] [14, 24, 75, 108, 127, 156, 166] [1, 15, 25, 92, 109, 150, 167] [16, 26, 71, 93, 110, 129, 158] [17, 27, 72, 94, 98, 111, 152] [14, 28, 48, 85, 95, 112, 131] [15, 29, 49, 96, 100, 113, 154] [3, 20, 30, 97, 114, 155, 161] [4, 21, 31, 102, 115, 120, 156] [22, 32, 76, 86, 99, 116, 157] [23, 33, 77, 104, 117, 122, 158] [20, 34, 54, 88, 101, 118, 159] [21, 35, 55, 85, 106, 119, 124] [6, 26, 36, 102, 120, 139, 164] [7, 27, 37, 103, 108, 121, 160] [28, 38, 79, 89, 104, 122, 141] [29, 39, 80, 105, 110, 123, 162] [26, 40, 60, 91, 106, 124, 143] [27, 41, 61, 87, 107, 112, 125] [11, 32, 42, 92, 109, 126, 163] [12, 33, 43, 114, 127, 132, 164] [34, 44, 82, 94, 111, 128, 165] [8, 35, 45, 89, 116, 129, 134] [32, 46, 66, 90, 96, 113, 130] [33, 47, 67, 91, 118, 131, 136] [16, 38, 48, 97, 114, 132, 151] [17, 39, 49, 115, 120, 133, 166] [40, 50, 83, 99, 116, 134, 153] [13, 41, 51, 93, 117, 122, 135] [38, 52, 71, 84, 101, 118, 136] [39, 53, 72, 95, 119, 124, 137] [22, 44, 54, 103, 121, 138, 167] [23, 45, 55, 97, 126, 139, 144] [18, 46, 56, 98, 105, 123, 140] [19, 47, 57, 99, 128, 141, 146] [44, 58, 76, 100, 107, 125, 142] [45, 59, 77, 101, 130, 143, 148] [28, 50, 60, 109, 126, 144, 161] [29, 51, 61, 102, 127, 132, 145] [24, 52, 62, 86, 111, 128, 146] [25, 53, 63, 104, 129, 134, 147] [50, 64, 79, 88, 113, 130, 148] [51, 65, 80, 106, 131, 136, 149] [34, 56, 66, 108, 115, 133, 150] [35, 57, 67, 109, 138, 151, 155] [30, 58, 68, 110, 117, 135, 152] [31, 59, 69, 111, 140, 153, 157] [56, 70, 82, 112, 119, 137, 154] [40, 62, 71, 92, 121, 138, 155] [41, 63, 72, 114, 139, 144, 156] [36, 64, 73, 94, 123, 140, 157] [37, 65, 74, 116, 141, 146, 158] [62, 75, 83, 96, 125, 142, 159] [46, 68, 76, 120, 127, 145, 160] [47, 69, 77, 121, 150, 161, 163] [42, 70, 78, 122, 129, 147, 162] [52, 73, 79, 103, 133, 150, 163] [53, 74, 80, 126, 151, 155, 164] [48, 75, 81, 105, 135, 152, 165] [58, 78, 82, 132, 139, 156, 166] [64, 81, 83, 115, 145, 160, 167]

H_Z (84 checks, sparse supports)

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

Code ID 168-18-10 · download JSON · raw on GitHub