[[74,2,11]] · QEC Challenge

[[74,2,11]] d ≤

kd²/n

3.27

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[4, 5, 7, 19, 29, 38, 44, 56, 63, 64, 67]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[3, 12, 14, 21, 34, 35, 37, 48, 58, 71, 73]

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 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)×37 (2,2)×74 (3,2)×185 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 37 (1,4): 37 (2,2): 74 (2,4): 222 (2,6): 222 (3,2): 185 (3,4): 1073 (3,6): 2627 (3,8): 1554 (3,10): 148

trapping sets H_Z (1,2)×37 (2,2)×74 (3,2)×185 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 37 (1,4): 37 (2,2): 74 (2,4): 222 (2,6): 222 (3,2): 185 (3,4): 1073 (3,6): 2627 (3,8): 1554 (3,10): 148

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(37,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order37_k2.txt, line 1. nonidentity supports a=[2], b=[2, 5, 15].

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[74,2,11]] — two-block group-algebra code on SmallGroup(37,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.27.

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_order37_k2.txt, line 1. This row is SmallGroup(37,1) with nonidentity GAP supports a=[2], b=[2, 5, 15]. 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 ≤ 11, not an exact certificate. The published distance (d=11) 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(37,1), 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], b=[2, 5, 15] 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 = 74 - 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 37 (max weight 6) · Z-checks 37 (max weight 6)

H_X (37 checks, sparse supports)

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

H_Z (37 checks, sparse supports)

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

Code ID 74-2-11 · download JSON · raw on GitHub