[[180,54,2]] · QEC Challenge

[[180,54,2]] d =

180

kd²/n

1.2

X/Z

Share this result

Distance

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

witness operator (support, 2 qubits)

[126, 135]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[75, 81]

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 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 (2,0)×90 (3,0)×72 (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 (2,0): 90 (2,2): 72 (2,4): 576 (2,6): 360 (3,0): 72 (3,2): 324 (3,4): 2700 (3,6): 4752 (3,8): 2592

trapping sets H_Z (1,2)×90 (2,0)×45 (3,2)×576 (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 (2,0): 45 (2,2): 72 (2,4): 846 (3,2): 576 (3,4): 4158 (3,6): 2052 (3,8): 180

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(90,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order90_k54.txt, line 1. nonidentity supports a=[2], b=[2, 11, 17].

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

[[180,54,2]] — two-block group-algebra code on SmallGroup(90,2)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.20.

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_k54.txt, line 1. This row is SmallGroup(90,2) with nonidentity GAP supports a=[2], b=[2, 11, 17]. 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=54), 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(90,2), 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=[2], b=[2, 11, 17] 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) = 54. 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, 1, 90, 91, 152, 178] [0, 1, 90, 91, 144, 179] [2, 5, 92, 95, 97, 134] [3, 6, 93, 96, 119, 162] [4, 7, 94, 126, 135, 143] [2, 5, 92, 94, 95, 143] [3, 6, 93, 96, 111, 168] [4, 7, 97, 126, 134, 135] [8, 14, 98, 104, 106, 149] [9, 15, 99, 105, 108, 151] [10, 16, 90, 91, 100, 157] [11, 17, 101, 107, 136, 173] [12, 18, 98, 102, 104, 159] [13, 19, 103, 125, 144, 179] [8, 14, 98, 100, 104, 157] [9, 15, 99, 102, 105, 159] [10, 16, 90, 91, 106, 149] [11, 17, 101, 107, 127, 176] [12, 18, 98, 104, 108, 151] [13, 19, 109, 117, 152, 178] [20, 28, 110, 118, 121, 164] [21, 29, 92, 95, 111, 168] [22, 30, 112, 120, 123, 165] [23, 31, 93, 96, 113, 170] [24, 32, 94, 114, 140, 143] [25, 33, 110, 115, 118, 171] [26, 34, 111, 116, 142, 168] [27, 35, 109, 117, 160, 175] [20, 28, 110, 113, 118, 170] [21, 29, 92, 95, 119, 162] [22, 30, 112, 115, 120, 171] [23, 31, 93, 96, 121, 164] [24, 32, 97, 122, 131, 134] [25, 33, 110, 118, 123, 165] [26, 34, 119, 124, 133, 162] [27, 35, 103, 125, 166, 172] [36, 45, 126, 135, 138, 174] [37, 46, 99, 105, 127, 176] [38, 47, 100, 128, 154, 157] [39, 48, 101, 107, 129, 177] [40, 49, 102, 130, 156, 159] [41, 50, 103, 122, 125, 131] [42, 51, 127, 132, 158, 176] [43, 52, 124, 128, 133, 154] [44, 53, 97, 134, 166, 172] [36, 45, 126, 129, 135, 177] [37, 46, 99, 105, 136, 173] [38, 47, 106, 137, 146, 149] [39, 48, 101, 107, 138, 174] [40, 49, 108, 139, 148, 151] [41, 50, 109, 114, 117, 140] [42, 51, 136, 141, 150, 173] [43, 52, 116, 137, 142, 146] [44, 53, 94, 143, 160, 175] [54, 62, 112, 120, 144, 179] [55, 63, 113, 145, 167, 170] [56, 64, 114, 137, 140, 146] [57, 65, 115, 147, 169, 171] [58, 66, 116, 139, 142, 148] [59, 67, 106, 109, 117, 149] [60, 68, 141, 145, 150, 167] [61, 69, 108, 137, 146, 151] [54, 62, 112, 120, 152, 178] [55, 63, 121, 153, 161, 164] [56, 64, 122, 128, 131, 154] [57, 65, 123, 155, 163, 165] [58, 66, 124, 130, 133, 156] [59, 67, 100, 103, 125, 157] [60, 68, 132, 153, 158, 161] [61, 69, 102, 128, 154, 159] [70, 76, 129, 160, 175, 177] [71, 77, 130, 153, 156, 161] [72, 78, 119, 122, 131, 162] [73, 79, 132, 155, 158, 163] [74, 80, 121, 124, 133, 164] [75, 81, 123, 153, 161, 165] [70, 76, 138, 166, 172, 174] [71, 77, 139, 145, 148, 167] [72, 78, 111, 114, 140, 168] [73, 79, 141, 147, 150, 169] [74, 80, 113, 116, 142, 170] [75, 81, 115, 145, 167, 171] [82, 85, 147, 166, 169, 172] [83, 86, 136, 139, 148, 173] [84, 87, 138, 141, 150, 174] [82, 85, 155, 160, 163, 175] [83, 86, 127, 130, 156, 176] [84, 87, 129, 132, 158, 177] [88, 89, 152, 155, 163, 178] [88, 89, 144, 147, 169, 179]

H_Z (90 checks, sparse supports)

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

Code ID 180-54-2 · download JSON · raw on GitHub