[[90,16,7]] · QEC Challenge

[[90,16,7]] d =

kd²/n

8.711

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[16, 34, 38, 64, 70, 72, 79]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[13, 22, 37, 42, 54, 58, 76]

certificate exact, d = 7 · scipy/HiGHS MILP

X: no logical < 7 exists; Z: no logical < 7 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): 45 (3,5): 630 (3,6): 4650 (3,7): 6390 (3,8): 2385 (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): 45 (3,5): 630 (3,6): 4650 (3,7): 6390 (3,8): 2385 (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

[[90,16,7]] — 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 = 8.71.

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_k16.txt, line 1. This row is SmallGroup(45,2) with nonidentity GAP supports a=[4, 24], b=[3, 22, 44]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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=[4, 24] and b=[3, 22, 44] 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) = 16. 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, 20, 26, 45, 52, 64, 87] [1, 9, 32, 46, 53, 57, 79] [2, 28, 34, 45, 47, 72, 89] [0, 3, 29, 48, 55, 60, 73] [4, 14, 37, 49, 58, 63, 84] [5, 16, 39, 46, 50, 60, 85] [1, 6, 17, 47, 51, 61, 66] [7, 35, 40, 47, 52, 56, 82] [2, 8, 36, 48, 53, 63, 80] [3, 9, 37, 54, 64, 69, 81] [10, 22, 42, 49, 55, 66, 88] [4, 11, 23, 50, 56, 67, 72] [12, 24, 43, 48, 50, 57, 71] [5, 13, 25, 51, 52, 58, 69] [6, 14, 26, 53, 59, 70, 75] [7, 15, 41, 49, 53, 60, 65] [8, 16, 42, 54, 61, 72, 86] [4, 9, 17, 62, 73, 78, 87] [18, 30, 44, 51, 55, 63, 77] [10, 19, 31, 56, 57, 64, 75] [11, 20, 32, 58, 65, 76, 80] [12, 21, 33, 45, 54, 58, 66] [13, 22, 34, 59, 60, 67, 78] [0, 14, 23, 61, 68, 79, 83] [15, 24, 44, 56, 61, 69, 74] [10, 16, 25, 62, 70, 80, 89] [11, 17, 26, 55, 71, 81, 85] [18, 27, 38, 46, 59, 64, 72] [19, 28, 39, 65, 66, 73, 83] [1, 20, 29, 67, 74, 84, 86] [21, 30, 40, 48, 62, 67, 75] [2, 22, 31, 68, 69, 76, 85] [3, 23, 32, 47, 70, 77, 88] [18, 24, 33, 65, 70, 78, 82] [19, 25, 34, 63, 71, 79, 86] [27, 35, 43, 51, 68, 73, 80] [5, 28, 36, 74, 75, 81, 88] [6, 29, 37, 50, 76, 82, 89] [7, 30, 38, 54, 71, 76, 83] [8, 31, 39, 52, 77, 78, 84] [27, 33, 40, 49, 74, 79, 85] [12, 35, 41, 59, 77, 81, 86] [13, 36, 42, 57, 82, 83, 87] [15, 38, 43, 45, 62, 84, 88] [21, 41, 44, 46, 68, 87, 89]

H_Z (45 checks, sparse supports)

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

Code ID 90-16-7 · download JSON · raw on GitHub