[[180,20,7]] · QEC Challenge

[[180,20,7]] d ≤

180

kd²/n

5.444

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[4, 11, 26, 27, 90, 135, 166]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[26, 62, 63, 76, 77, 135, 179]

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)×90 (2,2)×90 (3,2)×180 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 90 (1,4): 90 (2,2): 90 (2,4): 720 (2,6): 540 (3,2): 180 (3,4): 2700 (3,6): 8640 (3,8): 4050 (3,10): 360

trapping sets H_Z (1,2)×90 (2,2)×90 (3,2)×180 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 90 (1,4): 90 (2,2): 90 (2,4): 720 (2,6): 540 (3,2): 180 (3,4): 2700 (3,6): 8640 (3,8): 4050 (3,10): 360

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

[[180,20,7]] — two-block group-algebra code on SmallGroup(90,1)

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 = 5.44.

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_order90_k20.txt, line 5. This row is SmallGroup(90,1) with nonidentity GAP supports a=[4], b=[6, 12, 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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=[4] and b=[6, 12, 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 = 180 - rank(HX) - rank(HZ) = 20. 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 6) · Z-checks 90 (max weight 6)

H_X (90 checks, sparse supports)

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

H_Z (90 checks, sparse supports)

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

Code ID 180-20-7 · download JSON · raw on GitHub