[[180,8,17]] · QEC Challenge

[[180,8,17]] d ≤

180

kd²/n

12.844

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[22, 32, 38, 39, 51, 57, 64, 66, 70, 91, 94, 97, 122, 129, 145, 157, 169]

d_Z 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[26, 41, 47, 69, 77, 81, 82, 86, 103, 105, 114, 121, 131, 132, 137, 149, 177]

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)×90 (2,4)×315 (3,3)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 90 (1,4): 90 (2,4): 315 (2,5): 1080 (2,6): 450 (3,3): 90 (3,4): 180 (3,5): 1440 (3,6): 9270 (3,7): 12510 (3,8): 4320 (3,9): 1350 (3,10): 180

trapping sets H_Z (1,3)×90 (2,4)×315 (3,3)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 90 (1,4): 90 (2,4): 315 (2,5): 1080 (2,6): 450 (3,3): 90 (3,4): 180 (3,5): 1440 (3,6): 9270 (3,7): 12510 (3,8): 4320 (3,9): 1350 (3,10): 180

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

[[180,8,17]] — two-block group-algebra code on SmallGroup(90,1)

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 = 12.84.

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_order90_k8.txt, line 3. This row is SmallGroup(90,1) with nonidentity GAP supports a=[3, 38], b=[2, 10, 89]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 17, not an exact certificate. The published distance (d=17) 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(90,1), 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, 38] and b=[2, 10, 89] 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 = 180 - rank(HX) - rank(HZ) = 8. 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 90 (max weight 7) · Z-checks 90 (max weight 7)

H_X (90 checks, sparse supports)

[0, 36, 59, 90, 91, 133, 172] [1, 29, 45, 90, 91, 142, 175] [0, 2, 73, 92, 95, 116, 150] [3, 8, 54, 93, 124, 126, 151] [4, 55, 75, 94, 99, 109, 178] [1, 5, 46, 92, 95, 124, 158] [6, 48, 62, 96, 116, 135, 159] [7, 49, 63, 97, 103, 105, 179] [2, 8, 82, 98, 104, 132, 164] [3, 9, 20, 90, 99, 141, 165] [4, 10, 84, 100, 111, 123, 134] [11, 21, 70, 100, 101, 108, 144] [12, 22, 71, 102, 107, 113, 145] [13, 39, 72, 97, 103, 114, 160] [5, 14, 62, 98, 104, 141, 170] [6, 15, 64, 91, 105, 132, 171] [7, 16, 65, 106, 115, 119, 143] [17, 50, 76, 102, 106, 107, 152] [18, 67, 77, 101, 108, 121, 153] [19, 68, 78, 94, 109, 122, 166] [8, 20, 26, 110, 118, 149, 173] [9, 21, 36, 92, 111, 157, 174] [10, 22, 88, 112, 127, 140, 151] [11, 23, 37, 93, 112, 113, 122] [12, 24, 38, 94, 114, 121, 129] [13, 25, 56, 101, 106, 115, 130] [26, 40, 82, 96, 115, 116, 161] [27, 41, 83, 117, 132, 143, 162] [6, 14, 28, 110, 118, 157, 176] [15, 29, 76, 95, 119, 149, 177] [16, 30, 77, 120, 131, 136, 159] [17, 31, 66, 96, 114, 120, 121] [18, 32, 79, 97, 113, 122, 138] [19, 33, 80, 100, 107, 123, 139] [14, 34, 85, 93, 123, 124, 167] [35, 81, 86, 125, 134, 141, 168] [20, 36, 42, 117, 126, 135, 163] [0, 21, 37, 98, 127, 169, 178] [22, 38, 44, 128, 144, 156, 165] [23, 39, 54, 99, 128, 129, 139] [24, 40, 55, 100, 130, 138, 146] [25, 41, 70, 113, 120, 131, 147] [26, 42, 57, 102, 105, 131, 132] [27, 43, 58, 103, 133, 149, 159] [44, 60, 88, 92, 125, 134, 173] [15, 28, 45, 125, 126, 135, 169] [17, 29, 46, 104, 136, 163, 179] [18, 30, 47, 137, 148, 152, 171] [31, 48, 78, 105, 130, 137, 138] [32, 49, 85, 106, 129, 139, 154] [33, 50, 86, 112, 121, 140, 155] [28, 34, 51, 99, 108, 140, 141] [35, 52, 87, 109, 142, 151, 157] [30, 53, 89, 95, 117, 143, 176] [2, 37, 54, 110, 134, 144, 175] [38, 55, 61, 93, 145, 168, 174] [3, 39, 56, 111, 145, 146, 155] [4, 40, 57, 112, 147, 154, 160] [11, 41, 58, 129, 137, 148, 161] [42, 59, 71, 114, 119, 148, 149] [43, 60, 72, 115, 150, 163, 171] [44, 61, 74, 98, 117, 142, 151] [31, 46, 62, 118, 143, 152, 172] [32, 47, 63, 96, 153, 162, 177] [19, 48, 64, 119, 147, 153, 154] [34, 49, 65, 120, 146, 155, 166] [35, 50, 66, 128, 138, 156, 167] [45, 51, 67, 111, 122, 156, 157] [52, 68, 89, 123, 158, 165, 169] [47, 53, 69, 104, 125, 133, 159] [9, 56, 70, 94, 127, 160, 167] [10, 57, 71, 101, 128, 161, 166] [23, 58, 72, 102, 146, 153, 162] [12, 59, 73, 130, 136, 162, 163] [13, 60, 74, 131, 164, 172, 177] [61, 75, 83, 110, 133, 158, 165] [33, 64, 76, 97, 136, 161, 166] [51, 65, 77, 107, 137, 160, 167] [52, 66, 78, 108, 145, 154, 168] [1, 67, 79, 127, 139, 168, 169] [53, 68, 80, 140, 170, 174, 175] [63, 69, 81, 118, 142, 150, 171] [24, 73, 82, 103, 147, 152, 172] [25, 74, 83, 116, 148, 173, 179] [27, 75, 84, 126, 150, 170, 174] [5, 79, 85, 109, 144, 155, 175] [69, 80, 86, 124, 156, 176, 178] [7, 81, 87, 135, 158, 164, 177] [43, 84, 88, 90, 164, 176, 178] [16, 87, 89, 91, 170, 173, 179]

H_Z (90 checks, sparse supports)

[0, 1, 9, 88, 90, 92, 127] [0, 1, 15, 89, 91, 95, 169] [2, 5, 21, 44, 92, 98, 144] [3, 23, 34, 55, 93, 99, 146] [4, 19, 24, 70, 94, 100, 147] [2, 5, 29, 53, 95, 104, 175] [6, 26, 31, 63, 96, 105, 118] [7, 13, 32, 76, 97, 106, 177] [8, 14, 37, 61, 93, 98, 110] [4, 9, 39, 51, 99, 111, 160] [10, 11, 33, 40, 100, 112, 161] [11, 18, 25, 71, 101, 113, 148] [12, 17, 42, 72, 102, 114, 163] [7, 13, 43, 82, 103, 115, 164] [8, 14, 46, 69, 104, 118, 124] [7, 15, 42, 48, 105, 119, 135] [16, 17, 25, 49, 106, 120, 179] [12, 17, 33, 77, 107, 121, 136] [11, 18, 51, 78, 108, 122, 137] [4, 19, 52, 85, 109, 123, 154] [20, 28, 54, 75, 99, 110, 126] [10, 21, 56, 67, 101, 111, 127] [22, 23, 50, 57, 102, 112, 128] [12, 23, 32, 41, 113, 129, 162] [13, 24, 31, 59, 114, 130, 172] [16, 25, 26, 60, 115, 131, 173] [2, 6, 26, 83, 110, 116, 132] [27, 36, 53, 61, 117, 133, 174] [20, 28, 62, 81, 118, 135, 141] [16, 29, 59, 64, 91, 119, 136] [30, 31, 41, 65, 120, 137, 143] [18, 24, 31, 50, 121, 138, 152] [19, 23, 32, 67, 122, 139, 153] [10, 33, 34, 68, 123, 140, 166] [3, 5, 34, 86, 124, 141, 155] [35, 44, 45, 69, 125, 142, 156] [3, 36, 45, 84, 90, 111, 126] [22, 37, 70, 79, 113, 127, 144] [38, 39, 66, 71, 114, 128, 145] [24, 39, 49, 58, 103, 129, 146] [25, 40, 48, 73, 116, 130, 147] [30, 41, 42, 74, 117, 131, 148] [8, 15, 27, 42, 126, 132, 149] [0, 43, 69, 75, 133, 150, 178] [10, 35, 44, 54, 128, 134, 151] [6, 36, 45, 87, 91, 135, 157] [30, 46, 73, 76, 95, 136, 152] [47, 48, 58, 77, 137, 153, 159] [32, 40, 48, 66, 96, 138, 154] [33, 39, 49, 79, 97, 139, 155] [22, 50, 51, 80, 107, 140, 156] [9, 14, 35, 51, 141, 157, 167] [1, 52, 61, 81, 142, 158, 168] [16, 27, 53, 62, 143, 159, 170] [11, 38, 54, 85, 93, 129, 144] [12, 55, 56, 78, 94, 130, 145] [40, 56, 65, 72, 115, 146, 160] [41, 57, 64, 82, 132, 147, 161] [47, 58, 59, 83, 133, 148, 162] [20, 29, 43, 59, 90, 149, 163] [2, 60, 81, 84, 134, 150, 164] [3, 22, 52, 61, 145, 151, 165] [17, 47, 62, 82, 96, 104, 152] [18, 63, 64, 72, 97, 153, 171] [49, 57, 64, 78, 105, 154, 166] [50, 56, 65, 85, 106, 155, 167] [38, 66, 67, 86, 121, 156, 168] [21, 28, 52, 67, 108, 157, 169] [5, 68, 75, 87, 109, 158, 170] [6, 30, 43, 69, 159, 171, 176] [13, 57, 70, 77, 101, 131, 160] [26, 58, 71, 76, 102, 149, 161] [27, 63, 72, 73, 103, 150, 162] [36, 46, 60, 73, 92, 163, 172] [8, 74, 87, 88, 151, 164, 173] [9, 38, 68, 75, 94, 165, 174] [19, 65, 71, 76, 107, 119, 166] [34, 66, 70, 77, 108, 120, 167] [35, 55, 78, 79, 109, 138, 168] [37, 45, 68, 79, 122, 169, 175] [14, 80, 84, 89, 123, 170, 176] [15, 47, 60, 81, 125, 171, 177] [0, 62, 74, 82, 98, 116, 172] [20, 44, 83, 89, 117, 165, 173] [21, 55, 80, 84, 100, 174, 178] [1, 54, 80, 85, 124, 139, 175] [28, 53, 86, 88, 125, 140, 176] [29, 63, 74, 87, 142, 177, 179] [4, 37, 86, 88, 112, 134, 178] [7, 46, 83, 89, 143, 158, 179]

Code ID 180-8-17 · download JSON · raw on GitHub