[[76,2,12]] · QEC Challenge

[[76,2,12]] d ≤

kd²/n

3.789

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[2, 11, 18, 39, 40, 44, 46, 47, 48, 49, 51, 52]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[6, 9, 10, 17, 22, 24, 25, 49, 51, 56, 65, 69]

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)×38 (2,2)×38 (3,2)×76 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 38 (1,4): 38 (2,2): 38 (2,4): 304 (2,6): 228 (3,2): 76 (3,4): 1178 (3,6): 3496 (3,8): 1824 (3,10): 152

trapping sets H_Z (1,2)×38 (2,2)×38 (3,2)×76 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 38 (1,4): 38 (2,2): 38 (2,4): 304 (2,6): 228 (3,2): 76 (3,4): 1178 (3,6): 3496 (3,8): 1824 (3,10): 152

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

[[76,2,12]] — two-block group-algebra code on SmallGroup(38,2)

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 = 3.79.

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_order38_k2.txt, line 1. This row is SmallGroup(38,2) with nonidentity GAP supports a=[4], b=[5, 16, 23]. 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 ≤ 12, not an exact certificate. The published distance (d=12) 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(38,2), 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=[5, 16, 23] 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 = 76 - 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 38 (max weight 6) · Z-checks 38 (max weight 6)

H_X (38 checks, sparse supports)

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

H_Z (38 checks, sparse supports)

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

Code ID 76-2-12 · download JSON · raw on GitHub