[[84,6,11]] · QEC Challenge

[[84,6,11]] d ≤

kd²/n

8.643

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 7, w_Z = 7 (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)

[8, 22, 33, 45, 46, 50, 54, 60, 64, 66, 80]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[15, 38, 40, 41, 45, 48, 51, 53, 54, 61, 74]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×42 (2,4)×126 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 42 (3,4): 210 (3,5): 756 (3,6): 3864 (3,7): 5460 (3,8): 2142 (3,9): 756 (3,10): 168

trapping sets H_Z (1,3)×42 (2,4)×126 (3,3)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,4): 126 (2,5): 504 (2,6): 252 (3,3): 42 (3,4): 210 (3,5): 756 (3,6): 3864 (3,7): 5460 (3,8): 2142 (3,9): 756 (3,10): 168

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 ≤ 8 (computed)

How this code was found

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

[[84,6,11]] — two-block group-algebra code on SmallGroup(42,6)

Direction & hypothesis

Target: unrestricted weight-8 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 = 8.64.

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_wt7wtL3_order42_k6.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[4, 23], b=[6, 14, 27]. 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=6), 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

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(42,6), 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, 23] and b=[6, 14, 27] 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 = 84 - rank(HX) - rank(HZ) = 6. 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 42 (max weight 7) · Z-checks 42 (max weight 7)

H_X (42 checks, sparse supports)

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

H_Z (42 checks, sparse supports)

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

Code ID 84-6-11 · download JSON · raw on GitHub