[[180,60,2]] · QEC Challenge

[[180,60,2]] d =

180

kd²/n

1.333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[135, 153]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[37, 74]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 exists

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 (2,0)×135 (3,3)×570 (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 (2,0): 135 (2,5): 1080 (2,6): 360 (3,3): 570 (3,4): 1440 (3,6): 4860 (3,7): 10800 (3,8): 2160 (3,9): 1080

trapping sets H_Z (1,3)×90 (2,0)×90 (3,3)×30 (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 (2,0): 90 (2,4): 270 (2,5): 1080 (3,3): 30 (3,4): 1890 (3,5): 3240 (3,6): 4860 (3,7): 5940 (3,8): 180

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(90,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order90_k60.txt, line 1. nonidentity supports a=[5, 14], b=[2, 10, 16].

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 ≤ 8 (computed)

How this code was found

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

[[180,60,2]] — 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 = 1.33.

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_k60.txt, line 1. This row is SmallGroup(90,1) with nonidentity GAP supports a=[5, 14], b=[2, 10, 16]. 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=60), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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(90,1), 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=[5, 14], b=[2, 10, 16] 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) = 60. 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, 4, 13, 90, 91, 152, 172] [1, 7, 19, 90, 91, 144, 175] [2, 10, 25, 92, 95, 96, 116] [3, 12, 27, 93, 124, 126, 135] [0, 4, 13, 94, 109, 176, 178] [5, 16, 33, 92, 93, 95, 124] [6, 18, 35, 96, 116, 126, 135] [1, 7, 19, 97, 103, 173, 179] [8, 22, 41, 98, 104, 105, 132] [9, 24, 43, 90, 91, 99, 141] [2, 10, 25, 100, 123, 125, 134] [11, 26, 44, 101, 108, 144, 175] [3, 12, 27, 102, 107, 145, 168] [0, 4, 13, 97, 103, 160, 167] [14, 30, 50, 98, 99, 104, 141] [15, 32, 52, 90, 91, 105, 132] [5, 16, 33, 106, 115, 117, 143] [17, 34, 53, 102, 107, 152, 172] [6, 18, 35, 101, 108, 153, 162] [1, 7, 19, 94, 109, 161, 166] [20, 38, 58, 110, 118, 119, 149] [21, 40, 60, 92, 95, 111, 157] [8, 22, 41, 112, 140, 142, 151] [23, 42, 61, 93, 113, 122, 124] [9, 24, 43, 94, 109, 114, 121] [2, 10, 25, 101, 106, 108, 115] [11, 26, 44, 96, 116, 161, 166] [3, 12, 27, 117, 143, 153, 162] [28, 47, 66, 110, 111, 118, 157] [29, 49, 68, 92, 95, 119, 149] [14, 30, 50, 120, 131, 133, 159] [31, 51, 69, 96, 114, 116, 121] [15, 32, 52, 97, 103, 113, 122] [5, 16, 33, 100, 102, 107, 123] [17, 34, 53, 93, 124, 160, 167] [6, 18, 35, 125, 134, 145, 168] [36, 55, 72, 126, 135, 136, 163] [37, 57, 74, 98, 104, 127, 169] [20, 38, 58, 128, 156, 158, 165] [39, 59, 75, 99, 129, 139, 141] [21, 40, 60, 100, 123, 130, 138] [8, 22, 41, 113, 120, 122, 131] [23, 42, 61, 102, 105, 107, 132] [9, 24, 43, 97, 103, 133, 159] [11, 26, 44, 125, 134, 173, 179] [45, 63, 78, 126, 127, 135, 169] [46, 65, 80, 98, 104, 136, 163] [28, 47, 66, 137, 148, 150, 171] [48, 67, 81, 105, 130, 132, 138] [29, 49, 68, 106, 115, 129, 139] [14, 30, 50, 112, 114, 121, 140] [31, 51, 69, 99, 101, 108, 141] [15, 32, 52, 94, 109, 142, 151] [17, 34, 53, 117, 143, 176, 178] [54, 71, 83, 110, 118, 144, 175] [36, 55, 72, 145, 168, 170, 174] [56, 73, 84, 111, 146, 155, 157] [37, 57, 74, 112, 140, 147, 154] [20, 38, 58, 129, 137, 139, 148] [39, 59, 75, 114, 119, 121, 149] [21, 40, 60, 106, 115, 150, 171] [23, 42, 61, 117, 142, 143, 151] [62, 77, 86, 110, 118, 152, 172] [45, 63, 78, 153, 162, 164, 177] [64, 79, 87, 119, 147, 149, 154] [46, 65, 80, 120, 131, 146, 155] [28, 47, 66, 128, 130, 138, 156] [48, 67, 81, 111, 113, 122, 157] [29, 49, 68, 100, 123, 158, 165] [31, 51, 69, 125, 133, 134, 159] [70, 82, 88, 127, 160, 167, 169] [54, 71, 83, 128, 156, 161, 166] [36, 55, 72, 146, 153, 155, 162] [56, 73, 84, 130, 136, 138, 163] [37, 57, 74, 120, 131, 164, 177] [39, 59, 75, 133, 158, 159, 165] [76, 85, 89, 136, 161, 163, 166] [62, 77, 86, 137, 148, 160, 167] [45, 63, 78, 145, 147, 154, 168] [64, 79, 87, 127, 129, 139, 169] [46, 65, 80, 112, 140, 170, 174] [48, 67, 81, 142, 150, 151, 171] [70, 82, 88, 147, 152, 154, 172] [54, 71, 83, 137, 148, 173, 179] [56, 73, 84, 150, 170, 171, 174] [76, 85, 89, 144, 146, 155, 175] [62, 77, 86, 128, 156, 176, 178] [64, 79, 87, 158, 164, 165, 177] [70, 82, 88, 164, 176, 177, 178] [76, 85, 89, 170, 173, 174, 179]

H_Z (90 checks, sparse supports)

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

Code ID 180-60-2 · download JSON · raw on GitHub