[[92,4,13]] · QEC Challenge

[[92,4,13]] d ≤

kd²/n

7.348

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[5, 23, 24, 31, 32, 34, 46, 48, 49, 62, 65, 70, 77]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[3, 4, 7, 12, 34, 37, 39, 49, 51, 57, 58, 78, 88]

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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×46 (2,2)×46 (3,2)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 46 (1,6): 46 (2,2): 46 (2,4): 46 (2,6): 460 (2,8): 230 (2,10): 230 (3,2): 46 (3,4): 368 (3,6): 2714 (3,8): 4140 (3,10): 5658 (3,12): 3772 (3,14): 1150 (3,16): 46

trapping sets H_Z (1,2)×46 (2,2)×46 (3,2)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 46 (1,6): 46 (2,2): 46 (2,4): 46 (2,6): 460 (2,8): 230 (2,10): 230 (3,2): 46 (3,4): 368 (3,6): 2714 (3,8): 4140 (3,10): 5658 (3,12): 3772 (3,14): 1150 (3,16): 46

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(46,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order46_k4.txt, line 10. nonidentity supports a=[3], b=[2, 3, 8, 11, 37].

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

[[92,4,13]] — two-block group-algebra code on SmallGroup(46,1)

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 = 7.35.

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_wt8wtL2_order46_k4.txt, line 10. This row is SmallGroup(46,1) with nonidentity GAP supports a=[3], b=[2, 3, 8, 11, 37]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 13, 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(46,1), 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=[3], b=[2, 3, 8, 11, 37] 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 = 92 - rank(HX) - rank(HZ) = 4. 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 46 (max weight 8) · Z-checks 46 (max weight 8)

H_X (46 checks, sparse supports)

[0, 44, 46, 47, 53, 56, 82, 90] [1, 3, 46, 47, 52, 57, 83, 91] [0, 2, 46, 48, 51, 58, 84, 91] [3, 5, 47, 49, 50, 59, 85, 90] [2, 4, 48, 49, 50, 60, 86, 89] [5, 7, 48, 49, 51, 61, 87, 88] [4, 6, 47, 50, 52, 62, 87, 88] [7, 9, 46, 51, 53, 63, 86, 89] [6, 8, 52, 54, 64, 85, 90, 91] [9, 11, 53, 55, 65, 84, 90, 91] [8, 10, 46, 54, 56, 66, 83, 89] [11, 13, 47, 55, 57, 67, 82, 88] [10, 12, 48, 56, 58, 68, 81, 87] [13, 15, 49, 57, 59, 69, 80, 86] [12, 14, 50, 58, 60, 70, 79, 85] [15, 17, 51, 59, 61, 71, 78, 84] [14, 16, 52, 60, 62, 72, 77, 83] [17, 19, 53, 61, 63, 73, 76, 82] [16, 18, 54, 62, 64, 74, 75, 81] [19, 21, 55, 63, 65, 74, 75, 80] [18, 20, 56, 64, 66, 73, 76, 79] [21, 23, 57, 65, 67, 72, 77, 78] [20, 22, 58, 66, 68, 71, 77, 78] [23, 25, 59, 67, 69, 70, 76, 79] [22, 24, 60, 68, 69, 70, 75, 80] [25, 27, 61, 68, 69, 71, 74, 81] [24, 26, 62, 67, 70, 72, 73, 82] [27, 29, 63, 66, 71, 72, 73, 83] [26, 28, 64, 65, 71, 72, 74, 84] [29, 31, 64, 65, 70, 73, 75, 85] [28, 30, 63, 66, 69, 74, 76, 86] [31, 33, 62, 67, 68, 75, 77, 87] [30, 32, 61, 67, 68, 76, 78, 88] [33, 35, 60, 66, 69, 77, 79, 89] [32, 34, 59, 65, 70, 78, 80, 90] [35, 37, 58, 64, 71, 79, 81, 91] [34, 36, 46, 57, 63, 72, 80, 82] [37, 39, 47, 56, 62, 73, 81, 83] [36, 38, 48, 55, 61, 74, 82, 84] [39, 41, 49, 54, 60, 75, 83, 85] [38, 40, 50, 53, 59, 76, 84, 86] [41, 43, 51, 52, 58, 77, 85, 87] [40, 42, 51, 52, 57, 78, 86, 88] [43, 45, 50, 53, 56, 79, 87, 89] [42, 44, 49, 54, 55, 80, 88, 90] [1, 45, 48, 54, 55, 81, 89, 91]

H_Z (46 checks, sparse supports)

[0, 1, 2, 7, 10, 36, 46, 48] [0, 1, 3, 6, 11, 37, 47, 91] [2, 4, 5, 12, 38, 45, 48, 50] [3, 4, 5, 13, 39, 44, 47, 49] [3, 4, 6, 14, 40, 43, 50, 52] [2, 5, 7, 15, 41, 42, 49, 51] [1, 6, 8, 16, 41, 42, 52, 54] [0, 7, 9, 17, 40, 43, 51, 53] [8, 10, 18, 39, 44, 45, 54, 56] [9, 11, 19, 38, 44, 45, 53, 55] [0, 10, 12, 20, 37, 43, 56, 58] [1, 11, 13, 21, 36, 42, 55, 57] [2, 12, 14, 22, 35, 41, 58, 60] [3, 13, 15, 23, 34, 40, 57, 59] [4, 14, 16, 24, 33, 39, 60, 62] [5, 15, 17, 25, 32, 38, 59, 61] [6, 16, 18, 26, 31, 37, 62, 64] [7, 17, 19, 27, 30, 36, 61, 63] [8, 18, 20, 28, 29, 35, 64, 66] [9, 19, 21, 28, 29, 34, 63, 65] [10, 20, 22, 27, 30, 33, 66, 68] [11, 21, 23, 26, 31, 32, 65, 67] [12, 22, 24, 25, 31, 32, 68, 70] [13, 23, 24, 25, 30, 33, 67, 69] [14, 23, 24, 26, 29, 34, 70, 72] [15, 22, 25, 27, 28, 35, 69, 71] [16, 21, 26, 27, 28, 36, 72, 74] [17, 20, 26, 27, 29, 37, 71, 73] [18, 19, 25, 28, 30, 38, 74, 76] [18, 19, 24, 29, 31, 39, 73, 75] [17, 20, 23, 30, 32, 40, 76, 78] [16, 21, 22, 31, 33, 41, 75, 77] [15, 21, 22, 32, 34, 42, 78, 80] [14, 20, 23, 33, 35, 43, 77, 79] [13, 19, 24, 34, 36, 44, 80, 82] [12, 18, 25, 35, 37, 45, 79, 81] [0, 11, 17, 26, 36, 38, 82, 84] [1, 10, 16, 27, 37, 39, 81, 83] [2, 9, 15, 28, 38, 40, 84, 86] [3, 8, 14, 29, 39, 41, 83, 85] [4, 7, 13, 30, 40, 42, 86, 88] [5, 6, 12, 31, 41, 43, 85, 87] [5, 6, 11, 32, 42, 44, 88, 90] [4, 7, 10, 33, 43, 45, 87, 89] [0, 3, 8, 9, 34, 44, 46, 90] [1, 2, 8, 9, 35, 45, 89, 91]

Code ID 92-4-13 · download JSON · raw on GitHub