[[68,4,11]] · QEC Challenge

[[68,4,11]] d ≤

kd²/n

7.118

X/Z

Share this result

Distance

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

witness operator (support, 11 qubits)

[2, 4, 7, 9, 13, 15, 17, 25, 31, 58, 61]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[17, 25, 27, 31, 33, 38, 39, 45, 50, 51, 56]

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)×34 (2,2)×34 (3,2)×34 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 34 (1,6): 34 (2,2): 34 (2,4): 34 (2,6): 340 (2,8): 85 (2,10): 340 (3,2): 34 (3,4): 306 (3,6): 1836 (3,8): 2312 (3,10): 6018 (3,12): 2754 (3,14): 1768 (3,16): 204

trapping sets H_Z (1,2)×34 (2,2)×34 (3,2)×34 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 34 (1,6): 34 (2,2): 34 (2,4): 34 (2,6): 340 (2,8): 85 (2,10): 340 (3,2): 34 (3,4): 306 (3,6): 1836 (3,8): 2312 (3,10): 6018 (3,12): 2754 (3,14): 1768 (3,16): 204

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(34,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order34_k4.txt, line 1. nonidentity supports a=[3], b=[2, 3, 7, 10, 25].

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

[[68,4,11]] — two-block group-algebra code on SmallGroup(34,2)

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 = 7.12.

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_wt8wtL2_order34_k4.txt, line 1. This row is SmallGroup(34,2) with nonidentity GAP supports a=[3], b=[2, 3, 7, 10, 25]. 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 ≤ 11, 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(34,2), 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=[3], b=[2, 3, 7, 10, 25] 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 = 68 - 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 34 (max weight 8) · Z-checks 34 (max weight 8)

H_X (34 checks, sparse supports)

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

H_Z (34 checks, sparse supports)

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

Code ID 68-4-11 · download JSON · raw on GitHub