[[176,66,2]] · QEC Challenge

[[176,66,2]] d =

176

kd²/n

1.5

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)

[161, 165]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[120, 133]

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)×88 (2,0)×88 (3,0)×176 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 88 (1,4): 88 (2,0): 88 (2,2): 176 (2,4): 352 (2,6): 352 (3,0): 176 (3,2): 264 (3,4): 2376 (3,6): 3168 (3,8): 2112

trapping sets H_Z (1,2)×88 (2,0)×44 (3,2)×1408 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 88 (1,4): 88 (2,0): 44 (2,2): 176 (2,4): 616 (3,2): 1408 (3,4): 2112 (3,6): 1320 (3,8): 176

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(88,9), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order88_k66.txt, line 1. nonidentity supports a=[2], b=[2, 9, 13].

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

[[176,66,2]] — two-block group-algebra code on SmallGroup(88,9)

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 = 1.50.

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_wt6wtL2_order88_k66.txt, line 1. This row is SmallGroup(88,9) with nonidentity GAP supports a=[2], b=[2, 9, 13]. 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=66), 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(88,9), 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, 9, 13] 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 = 176 - rank(HX) - rank(HZ) = 66. 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 88 (max weight 6) · Z-checks 88 (max weight 6)

H_X (88 checks, sparse supports)

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

H_Z (88 checks, sparse supports)

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

Code ID 176-66-2 · download JSON · raw on GitHub