[[180,2,15]] · QEC Challenge

[[180,2,15]] d ≤

180

kd²/n

2.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[33, 41, 56, 59, 77, 79, 94, 100, 106, 120, 139, 150, 158, 161, 168]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[2, 4, 5, 14, 17, 20, 33, 51, 57, 121, 129, 141, 160, 164, 168]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×90 (2,2)×90 (3,2)×90 (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 (1,3): 90 (2,2): 90 (2,3): 540 (2,4): 270 (3,2): 90 (3,3): 1620 (3,4): 2970 (3,5): 1350 (3,6): 540 (3,7): 90

trapping sets H_Z (1,2)×90 (2,2)×90 (3,2)×90 (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 (1,3): 90 (2,2): 90 (2,3): 540 (2,4): 270 (3,2): 90 (3,3): 1620 (3,4): 2970 (3,5): 1350 (3,6): 540 (3,7): 90

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(90,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order90_k2.txt, line 13. nonidentity supports a=[16], b=[18, 24].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

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

[[180,2,15]] — two-block group-algebra code on SmallGroup(90,6)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.50.

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_wt5wtL2_order90_k2.txt, line 13. This row is SmallGroup(90,6) with nonidentity GAP supports a=[16], b=[18, 24]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 15, not an exact certificate. The published distance (d=15) 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

DeepSeek V4 Flash 0731 (provenance.model), 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(90,6), 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=[16], b=[18, 24] 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) = 2. 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 5) · Z-checks 90 (max weight 5)

H_X (90 checks, sparse supports)

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

H_Z (90 checks, sparse supports)

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

Code ID 180-2-15 · download JSON · raw on GitHub