[[84,4,11]] · QEC Challenge

[[84,4,11]] d ≤

kd²/n

5.762

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)

[7, 12, 18, 20, 22, 41, 48, 51, 56, 68, 79]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[0, 1, 7, 18, 30, 47, 66, 68, 70, 74, 79]

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)×42 (2,2)×42 (3,2)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 42 (1,4): 42 (2,2): 42 (2,4): 336 (2,6): 252 (3,2): 42 (3,4): 1428 (3,6): 3822 (3,8): 2142 (3,10): 168

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

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

(1,2): 42 (1,4): 42 (2,2): 42 (2,4): 336 (2,6): 252 (3,2): 42 (3,4): 1428 (3,6): 3822 (3,8): 2142 (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 ≤ 6 (computed)

How this code was found

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

[[84,4,11]] — two-block group-algebra code on SmallGroup(42,6)

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 = 5.76.

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_order42_k4.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[8], b=[4, 12, 40]. 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 (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=[8] and b=[4, 12, 40] 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) = 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 42 (max weight 6) · Z-checks 42 (max weight 6)

H_X (42 checks, sparse supports)

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

H_Z (42 checks, sparse supports)

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

Code ID 84-4-11 · download JSON · raw on GitHub