[[90,8,11]] · QEC Challenge

[[90,8,11]] d ≤

kd²/n

10.756

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)

[1, 31, 42, 46, 47, 50, 57, 59, 63, 69, 70]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[9, 13, 26, 28, 30, 37, 45, 46, 74, 82, 89]

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)×45 (2,4)×135 (3,3)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 45 (1,4): 45 (2,4): 135 (2,5): 540 (2,6): 270 (3,3): 45 (3,4): 90 (3,5): 630 (3,6): 4635 (3,7): 6390 (3,8): 2025 (3,9): 810 (3,10): 180

trapping sets H_Z (1,3)×45 (2,4)×135 (3,3)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 45 (1,4): 45 (2,4): 135 (2,5): 540 (2,6): 270 (3,3): 45 (3,4): 90 (3,5): 630 (3,6): 4635 (3,7): 6390 (3,8): 2025 (3,9): 810 (3,10): 180

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

[[90,8,11]] — two-block group-algebra code on SmallGroup(45,1)

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 = 10.76.

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_wt7wtL3_order45_k8.txt, line 1. This row is SmallGroup(45,1) with nonidentity GAP supports a=[6, 17], b=[2, 24, 31]. 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=8), 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(45,1), 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=[6, 17] and b=[2, 24, 31] 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 = 90 - rank(HX) - rank(HZ) = 8. 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 45 (max weight 7) · Z-checks 45 (max weight 7)

H_X (45 checks, sparse supports)

[0, 34, 44, 45, 65, 74, 80] [1, 24, 39, 45, 46, 47, 78] [2, 20, 40, 47, 49, 74, 82] [3, 15, 35, 48, 49, 55, 86] [4, 30, 42, 46, 49, 50, 83] [0, 5, 43, 47, 48, 50, 52] [6, 21, 33, 48, 51, 53, 85] [7, 9, 29, 52, 55, 82, 87] [4, 8, 24, 53, 55, 56, 63] [9, 25, 41, 54, 56, 64, 89] [1, 10, 44, 50, 51, 55, 57] [11, 27, 38, 51, 56, 58, 88] [2, 12, 14, 52, 53, 57, 60] [3, 13, 30, 53, 54, 58, 61] [14, 31, 40, 54, 59, 62, 69] [15, 17, 37, 60, 63, 87, 89] [0, 10, 16, 61, 63, 64, 72] [11, 17, 33, 62, 64, 65, 73] [5, 18, 20, 57, 58, 63, 66] [6, 19, 35, 58, 59, 64, 67] [20, 36, 43, 59, 65, 68, 75] [7, 21, 23, 60, 61, 66, 69] [1, 8, 22, 61, 62, 67, 70] [9, 23, 38, 45, 62, 68, 71] [24, 26, 42, 65, 69, 72, 89] [2, 18, 25, 70, 72, 73, 80] [3, 19, 26, 71, 73, 74, 81] [12, 27, 29, 66, 67, 72, 75] [4, 13, 28, 67, 68, 73, 76] [14, 29, 41, 46, 68, 74, 77] [15, 30, 32, 45, 69, 70, 75] [5, 16, 31, 70, 71, 76, 78] [6, 17, 32, 47, 71, 77, 79] [7, 27, 33, 49, 78, 80, 81] [8, 28, 34, 79, 81, 82, 86] [21, 35, 37, 46, 75, 76, 80] [10, 22, 36, 76, 77, 81, 83] [11, 23, 37, 50, 77, 82, 84] [12, 25, 38, 48, 78, 79, 83] [13, 26, 39, 52, 79, 84, 85] [16, 36, 40, 56, 85, 86, 87] [18, 31, 41, 51, 83, 84, 86] [19, 32, 42, 57, 84, 87, 88] [22, 34, 43, 54, 60, 85, 88] [28, 39, 44, 59, 66, 88, 89]

H_Z (45 checks, sparse supports)

[0, 1, 23, 30, 45, 50, 61] [1, 4, 29, 35, 46, 55, 67] [1, 2, 5, 32, 47, 57, 70] [3, 5, 6, 38, 48, 58, 71] [2, 3, 4, 33, 49, 53, 73] [4, 5, 10, 37, 50, 63, 76] [6, 10, 11, 41, 51, 64, 77] [5, 7, 12, 39, 52, 66, 78] [6, 8, 12, 13, 53, 67, 79] [9, 13, 14, 43, 52, 54, 68] [3, 7, 8, 10, 55, 61, 81] [8, 9, 11, 40, 56, 62, 82] [10, 12, 18, 42, 57, 72, 83] [11, 13, 18, 19, 58, 73, 84] [14, 19, 20, 44, 57, 59, 74] [12, 15, 21, 43, 48, 60, 75] [13, 16, 21, 22, 61, 76, 85] [14, 17, 22, 23, 60, 62, 77] [8, 15, 16, 18, 63, 70, 86] [9, 16, 17, 19, 64, 71, 87] [0, 17, 20, 24, 47, 63, 65] [18, 21, 27, 44, 51, 66, 80] [19, 22, 27, 28, 67, 81, 88] [20, 23, 28, 29, 66, 68, 82] [14, 21, 24, 30, 46, 53, 69] [22, 25, 30, 31, 54, 70, 83] [23, 26, 31, 32, 69, 71, 84] [16, 24, 25, 27, 56, 72, 78] [17, 25, 26, 28, 73, 79, 89] [0, 2, 26, 29, 52, 72, 74] [20, 27, 30, 35, 49, 58, 75] [28, 31, 35, 36, 59, 76, 86] [29, 32, 36, 37, 75, 77, 87] [1, 31, 33, 38, 51, 62, 78] [32, 34, 38, 39, 45, 79, 88] [0, 25, 33, 35, 48, 64, 80] [26, 33, 34, 36, 65, 81, 85] [2, 7, 34, 37, 60, 80, 82] [4, 36, 38, 41, 56, 68, 83] [37, 39, 41, 42, 46, 84, 89] [6, 39, 40, 43, 47, 59, 85] [3, 34, 40, 41, 54, 74, 86] [7, 15, 40, 42, 49, 69, 87] [11, 42, 43, 44, 50, 65, 88] [9, 15, 24, 44, 45, 55, 89]

Code ID 90-8-11 · download JSON · raw on GitHub