[[156,4,18]] · QEC Challenge

[[156,4,18]] d ≤

156

kd²/n

8.308

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[30, 45, 54, 57, 69, 72, 74, 77, 81, 85, 98, 109, 114, 120, 128, 139, 144, 150]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[11, 19, 23, 32, 33, 47, 57, 75, 97, 102, 105, 117, 120, 132, 135, 144, 153, 155]

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 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)×78 (2,4)×234 (3,3)×78 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 78 (1,4): 78 (2,4): 234 (2,5): 936 (2,6): 468 (3,3): 78 (3,4): 312 (3,5): 702 (3,6): 7488 (3,7): 12246 (3,8): 3744 (3,9): 1404 (3,10): 312

trapping sets H_Z (1,3)×78 (2,4)×234 (3,3)×78 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 78 (1,4): 78 (2,4): 234 (2,5): 936 (2,6): 468 (3,3): 78 (3,4): 312 (3,5): 702 (3,6): 7488 (3,7): 12246 (3,8): 3744 (3,9): 1404 (3,10): 312

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

[[156,4,18]] — two-block group-algebra code on SmallGroup(78,2)

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 = 8.31.

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_order78_k4.txt, line 12. This row is SmallGroup(78,2) with nonidentity GAP supports a=[3, 13], b=[4, 15, 30]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, not an exact certificate. The published distance (d=18) 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(78,2), 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, 13] and b=[4, 15, 30] 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 = 156 - rank(HX) - rank(HZ) = 4. 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 78 (max weight 7) · Z-checks 78 (max weight 7)

H_X (78 checks, sparse supports)

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

H_Z (78 checks, sparse supports)

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

Code ID 156-4-18 · download JSON · raw on GitHub