[[176,46,4]] · QEC Challenge

[[176,46,4]] d ≤

176

kd²/n

4.182

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[153, 157, 160, 164]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[41, 45, 48, 52]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×176 (2,2)×88 (3,4)×2552 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 176 (2,2): 88 (2,4): 616 (2,6): 968 (3,4): 2552 (3,6): 9328 (3,8): 8448

trapping sets H_Z (1,4)×176 (2,2)×88 (3,4)×2552 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 176 (2,2): 88 (2,4): 616 (2,6): 968 (3,4): 2552 (3,6): 9328 (3,8): 8448

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(88,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order88_k46.txt, line 1. nonidentity supports a=[2, 3, 6], b=[3, 5, 10].

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

[[176,46,4]] — two-block group-algebra code on SmallGroup(88,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 4.18.

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_wt8wtL4_order88_k46.txt, line 1. This row is SmallGroup(88,1) with nonidentity GAP supports a=[2, 3, 6], b=[3, 5, 10]. 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=46), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance 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(88,1), 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=[2, 3, 6], b=[3, 5, 10] 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) = 46. 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 8) · Z-checks 88 (max weight 8)

H_X (88 checks, sparse supports)

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

H_Z (88 checks, sparse supports)

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

Code ID 176-46-4 · download JSON · raw on GitHub