[[90,28,2]] · QEC Challenge

[[90,28,2]] d =

kd²/n

1.244

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[68, 74]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[32, 37]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 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 4 · H_Z 4 (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 (2,0)×45 (3,0)×15 (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 (2,0): 45 (2,4): 90 (2,5): 540 (2,6): 90 (3,0): 15 (3,3): 15 (3,4): 675 (3,5): 1080 (3,6): 2790 (3,7): 4320 (3,8): 180 (3,9): 270

trapping sets H_Z (1,3)×45 (2,0)×45 (3,0)×15 (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 (2,0): 45 (2,4): 90 (2,5): 540 (2,6): 90 (3,0): 15 (3,3): 15 (3,4): 675 (3,5): 1080 (3,6): 2790 (3,7): 4320 (3,8): 180 (3,9): 270

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(45,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order45_k28.txt, line 1. nonidentity supports a=[2, 5], b=[3, 4, 9].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

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

[[90,28,2]] — 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 = 1.24.

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_k28.txt, line 1. This row is SmallGroup(45,2) with nonidentity GAP supports a=[2, 5], b=[3, 4, 9]. 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=28), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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

DeepSeek V4 Flash 0731 (provenance.model), 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=[2, 5], b=[3, 4, 9] 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) = 28. 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, 1, 4, 45, 52, 71, 85] [0, 1, 4, 46, 57, 77, 88] [2, 5, 10, 45, 47, 71, 79] [3, 6, 11, 45, 48, 52, 60] [0, 1, 4, 49, 63, 82, 89] [2, 5, 10, 46, 50, 77, 84] [3, 6, 11, 46, 51, 57, 66] [7, 12, 18, 47, 52, 79, 85] [8, 13, 19, 45, 47, 48, 53] [9, 14, 20, 48, 54, 60, 69] [2, 5, 10, 49, 55, 82, 87] [3, 6, 11, 49, 56, 63, 72] [7, 12, 18, 50, 57, 84, 88] [8, 13, 19, 46, 50, 51, 58] [9, 14, 20, 51, 59, 66, 75] [15, 21, 27, 47, 52, 53, 60] [16, 22, 28, 48, 53, 54, 61] [17, 23, 29, 54, 62, 69, 78] [7, 12, 18, 55, 63, 87, 89] [8, 13, 19, 49, 55, 56, 64] [9, 14, 20, 56, 65, 72, 80] [15, 21, 27, 50, 57, 58, 66] [16, 22, 28, 51, 58, 59, 67] [17, 23, 29, 59, 68, 75, 83] [24, 30, 35, 53, 60, 61, 69] [25, 31, 36, 54, 61, 62, 70] [26, 32, 37, 62, 71, 78, 85] [15, 21, 27, 55, 63, 64, 72] [16, 22, 28, 56, 64, 65, 73] [17, 23, 29, 65, 74, 80, 86] [24, 30, 35, 58, 66, 67, 75] [25, 31, 36, 59, 67, 68, 76] [26, 32, 37, 68, 77, 83, 88] [33, 38, 41, 61, 69, 70, 78] [34, 39, 42, 62, 70, 71, 79] [24, 30, 35, 64, 72, 73, 80] [25, 31, 36, 65, 73, 74, 81] [26, 32, 37, 74, 82, 86, 89] [33, 38, 41, 67, 75, 76, 83] [34, 39, 42, 68, 76, 77, 84] [40, 43, 44, 70, 78, 79, 85] [33, 38, 41, 73, 80, 81, 86] [34, 39, 42, 74, 81, 82, 87] [40, 43, 44, 76, 83, 84, 88] [40, 43, 44, 81, 86, 87, 89]

H_Z (45 checks, sparse supports)

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

Code ID 90-28-2 · download JSON · raw on GitHub