[[94,2,13]] · QEC Challenge

[[94,2,13]] d ≤

kd²/n

3.596

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)

[11, 13, 15, 32, 45, 54, 55, 57, 62, 78, 79, 90, 92]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[2, 16, 18, 22, 35, 38, 44, 47, 52, 64, 83, 86, 92]

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)×47 (2,2)×47 (3,2)×94 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 47 (1,4): 47 (2,2): 47 (2,4): 376 (2,6): 282 (3,2): 94 (3,4): 1457 (3,6): 4324 (3,8): 2256 (3,10): 188

trapping sets H_Z (1,2)×47 (2,2)×47 (3,2)×94 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 47 (1,4): 47 (2,2): 47 (2,4): 376 (2,6): 282 (3,2): 94 (3,4): 1457 (3,6): 4324 (3,8): 2256 (3,10): 188

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

[[94,2,13]] — two-block group-algebra code on SmallGroup(47,1)

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 = 2·169/94 = 3.60.

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_order47_k2.txt, line 1. This row is SmallGroup(47,1) with nonidentity GAP supports a=[2], b=[3,6,19]. 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(47,1), 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] and b=[3,6,19] 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 = 94 - 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 47 (max weight 6) · Z-checks 47 (max weight 6)

H_X (47 checks, sparse supports)

[0, 46, 47, 76, 89, 92] [0, 1, 48, 77, 90, 93] [1, 2, 47, 49, 78, 91] [2, 3, 48, 50, 79, 92] [3, 4, 49, 51, 80, 93] [4, 5, 47, 50, 52, 81] [5, 6, 48, 51, 53, 82] [6, 7, 49, 52, 54, 83] [7, 8, 50, 53, 55, 84] [8, 9, 51, 54, 56, 85] [9, 10, 52, 55, 57, 86] [10, 11, 53, 56, 58, 87] [11, 12, 54, 57, 59, 88] [12, 13, 55, 58, 60, 89] [13, 14, 56, 59, 61, 90] [14, 15, 57, 60, 62, 91] [15, 16, 58, 61, 63, 92] [16, 17, 59, 62, 64, 93] [17, 18, 47, 60, 63, 65] [18, 19, 48, 61, 64, 66] [19, 20, 49, 62, 65, 67] [20, 21, 50, 63, 66, 68] [21, 22, 51, 64, 67, 69] [22, 23, 52, 65, 68, 70] [23, 24, 53, 66, 69, 71] [24, 25, 54, 67, 70, 72] [25, 26, 55, 68, 71, 73] [26, 27, 56, 69, 72, 74] [27, 28, 57, 70, 73, 75] [28, 29, 58, 71, 74, 76] [29, 30, 59, 72, 75, 77] [30, 31, 60, 73, 76, 78] [31, 32, 61, 74, 77, 79] [32, 33, 62, 75, 78, 80] [33, 34, 63, 76, 79, 81] [34, 35, 64, 77, 80, 82] [35, 36, 65, 78, 81, 83] [36, 37, 66, 79, 82, 84] [37, 38, 67, 80, 83, 85] [38, 39, 68, 81, 84, 86] [39, 40, 69, 82, 85, 87] [40, 41, 70, 83, 86, 88] [41, 42, 71, 84, 87, 89] [42, 43, 72, 85, 88, 90] [43, 44, 73, 86, 89, 91] [44, 45, 74, 87, 90, 92] [45, 46, 75, 88, 91, 93]

H_Z (47 checks, sparse supports)

[0, 2, 5, 18, 47, 48] [1, 3, 6, 19, 48, 49] [2, 4, 7, 20, 49, 50] [3, 5, 8, 21, 50, 51] [4, 6, 9, 22, 51, 52] [5, 7, 10, 23, 52, 53] [6, 8, 11, 24, 53, 54] [7, 9, 12, 25, 54, 55] [8, 10, 13, 26, 55, 56] [9, 11, 14, 27, 56, 57] [10, 12, 15, 28, 57, 58] [11, 13, 16, 29, 58, 59] [12, 14, 17, 30, 59, 60] [13, 15, 18, 31, 60, 61] [14, 16, 19, 32, 61, 62] [15, 17, 20, 33, 62, 63] [16, 18, 21, 34, 63, 64] [17, 19, 22, 35, 64, 65] [18, 20, 23, 36, 65, 66] [19, 21, 24, 37, 66, 67] [20, 22, 25, 38, 67, 68] [21, 23, 26, 39, 68, 69] [22, 24, 27, 40, 69, 70] [23, 25, 28, 41, 70, 71] [24, 26, 29, 42, 71, 72] [25, 27, 30, 43, 72, 73] [26, 28, 31, 44, 73, 74] [27, 29, 32, 45, 74, 75] [28, 30, 33, 46, 75, 76] [0, 29, 31, 34, 76, 77] [1, 30, 32, 35, 77, 78] [2, 31, 33, 36, 78, 79] [3, 32, 34, 37, 79, 80] [4, 33, 35, 38, 80, 81] [5, 34, 36, 39, 81, 82] [6, 35, 37, 40, 82, 83] [7, 36, 38, 41, 83, 84] [8, 37, 39, 42, 84, 85] [9, 38, 40, 43, 85, 86] [10, 39, 41, 44, 86, 87] [11, 40, 42, 45, 87, 88] [12, 41, 43, 46, 88, 89] [0, 13, 42, 44, 89, 90] [1, 14, 43, 45, 90, 91] [2, 15, 44, 46, 91, 92] [0, 3, 16, 45, 92, 93] [1, 4, 17, 46, 47, 93]

Code ID 94-2-13 · download JSON · raw on GitHub