[[92,2,13]] · QEC Challenge

[[92,2,13]] d ≤

kd²/n

3.674

X/Z

Share this result

Distance

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

witness operator (support, 13 qubits)

[13, 26, 41, 47, 48, 60, 63, 75, 76, 85, 86, 89, 90]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[3, 8, 9, 11, 12, 16, 37, 43, 50, 54, 59, 72, 84]

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 6 · H_Z 6 (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)×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 (1,4): 46 (2,2): 46 (2,4): 368 (2,6): 276 (3,2): 46 (3,4): 1656 (3,6): 3910 (3,8): 2346 (3,10): 184

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 (1,4): 46 (2,2): 46 (2,4): 368 (2,6): 276 (3,2): 46 (3,4): 1656 (3,6): 3910 (3,8): 2346 (3,10): 184

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[92,2,13]] — two-block group-algebra code on SmallGroup(46,2)

Direction & hypothesis

Target: unrestricted weight-6 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 = 3.67.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order46_k2.txt, line 1. This row is SmallGroup(46,2) with nonidentity GAP supports a=[4], b=[6, 12, 19]. 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=2), 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 (d=13) 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

Union Alpha (maker currently anonymous), 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(46,2), 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=[4] and b=[6, 12, 19] 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) = 2. 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 6) · Z-checks 46 (max weight 6)

H_X (46 checks, sparse supports)

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

H_Z (46 checks, sparse supports)

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

Code ID 92-2-13 · download JSON · raw on GitHub