[[90,4,12]] · QEC Challenge

[[90,4,12]] d ≤

kd²/n

6.4

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[6, 12, 14, 26, 44, 59, 62, 69, 70, 83, 86, 87]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[2, 7, 9, 12, 17, 23, 38, 41, 47, 49, 72, 82]

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 3 · H_Z 3

trapping sets H_X (1,3)×90 (2,4)×675 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 90 (2,4): 675 (3,3): 135 (3,5): 6345 (3,7): 900

trapping sets H_Z (1,3)×90 (2,4)×675 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 90 (2,4): 675 (3,3): 135 (3,5): 6345 (3,7): 900

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

[[90,4,12]] — two-block group-algebra code on SmallGroup(45,1)

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 = 6.40.

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_wt6wtL3_order45_k4.txt, line 1. This row is SmallGroup(45,1) with nonidentity GAP supports a=[2, 11], b=[14, 36]. 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 ≤ 12, not an exact certificate. The published distance (d=12) 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=[2, 11] and b=[14, 36] 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) = 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 45 (max weight 6) · Z-checks 45 (max weight 6)

H_X (45 checks, sparse supports)

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

H_Z (45 checks, sparse supports)

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

Code ID 90-4-12 · download JSON · raw on GitHub