[[90,12,9]] · QEC Challenge

[[90,12,9]] d =

kd²/n

10.8

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[20, 47, 48, 49, 57, 59, 77, 86, 87]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[13, 24, 27, 28, 57, 61, 66, 74, 82]

certificate exact, d = 9 · CryptoMiniSat 5.14 SAT

X: no logical < 9 exists; Z: no logical < 9 exists

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): 135 (3,5): 720 (3,6): 4515 (3,7): 6120 (3,8): 1980 (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): 135 (3,5): 720 (3,6): 4515 (3,7): 6120 (3,8): 1980 (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

[[90,12,9]] — two-block group-algebra code on SmallGroup(45,2)

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 = 10.80.

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_k12.txt, line 2. This row is SmallGroup(45,2) with nonidentity GAP supports a=[7, 17], b=[2, 16, 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 9, not an exact certificate. The published distance (d=9) 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(45,2), 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=[7, 17] and b=[2, 16, 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) = 12. 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, 33, 37, 45, 49, 76, 79] [1, 26, 38, 45, 46, 81, 84] [2, 17, 42, 47, 55, 83, 85] [3, 4, 40, 47, 48, 56, 84] [4, 32, 41, 46, 49, 70, 87] [5, 23, 34, 47, 50, 86, 88] [0, 6, 43, 48, 50, 51, 87] [7, 25, 44, 52, 63, 68, 71] [8, 10, 26, 52, 53, 64, 88] [7, 9, 11, 50, 53, 54, 65] [10, 29, 39, 50, 55, 78, 89] [1, 11, 44, 51, 55, 56, 79] [12, 31, 40, 52, 57, 74, 77] [2, 13, 32, 53, 57, 58, 89] [3, 12, 14, 54, 55, 58, 59] [15, 18, 34, 45, 60, 72, 77] [0, 16, 19, 57, 60, 61, 73] [15, 17, 20, 58, 61, 62, 74] [18, 36, 43, 57, 62, 63, 82] [5, 19, 37, 58, 63, 64, 85] [6, 18, 20, 47, 59, 64, 65] [7, 21, 39, 46, 60, 66, 82] [1, 8, 22, 61, 63, 66, 67] [9, 21, 23, 62, 64, 67, 68] [2, 24, 27, 46, 48, 69, 80] [3, 25, 28, 66, 69, 70, 81] [24, 26, 29, 67, 70, 71, 82] [12, 27, 42, 49, 66, 71, 72] [4, 13, 28, 52, 67, 72, 73] [14, 27, 29, 53, 68, 73, 74] [5, 15, 30, 49, 51, 69, 75] [6, 16, 31, 70, 72, 75, 76] [17, 30, 32, 71, 73, 76, 77] [8, 33, 35, 51, 54, 78, 86] [9, 34, 36, 75, 78, 79, 87] [10, 21, 35, 45, 56, 75, 80] [11, 22, 36, 60, 76, 80, 81] [23, 35, 37, 61, 77, 81, 82] [13, 24, 38, 56, 59, 78, 83] [14, 25, 39, 79, 80, 83, 84] [16, 40, 41, 59, 62, 85, 89] [19, 30, 41, 48, 65, 83, 86] [20, 31, 42, 69, 84, 86, 87] [22, 33, 43, 65, 68, 85, 88] [28, 38, 44, 54, 74, 88, 89]

H_Z (45 checks, sparse supports)

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

Code ID 90-12-9 · download JSON · raw on GitHub