[[90,10,7]] · QEC Challenge

[[90,10,7]] d =

kd²/n

5.444

X/Z

Share this result

Distance

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

witness operator (support, 7 qubits)

[7, 18, 22, 26, 64, 68, 77]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[25, 33, 41, 66, 76, 79, 87]

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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×45 (2,2)×45 (3,2)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 45 (1,4): 45 (2,2): 45 (2,4): 360 (2,6): 270 (3,2): 90 (3,4): 1350 (3,6): 4320 (3,8): 2025 (3,10): 180

trapping sets H_Z (1,2)×45 (2,2)×45 (3,2)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 45 (1,4): 45 (2,2): 45 (2,4): 360 (2,6): 270 (3,2): 90 (3,4): 1350 (3,6): 4320 (3,8): 2025 (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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[90,10,7]] — 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 = 5.44.

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_wt6wtL2_order45_k10.txt, line 1. This row is SmallGroup(45,1) with nonidentity GAP supports a=[2], b=[3, 5, 14]. 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=10), 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(45,1), 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] and b=[3, 5, 14] 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) = 10. 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, 45, 59, 69, 86] [0, 1, 46, 65, 75, 85] [2, 29, 45, 47, 56, 68] [3, 4, 46, 48, 78, 89] [1, 4, 45, 49, 80, 88] [2, 5, 46, 50, 54, 74] [3, 6, 49, 51, 69, 83] [7, 37, 47, 52, 64, 77] [8, 10, 48, 50, 53, 65] [9, 11, 51, 54, 80, 85] [5, 10, 47, 49, 55, 59] [6, 11, 48, 56, 75, 86] [7, 12, 50, 57, 62, 82] [8, 13, 45, 51, 55, 58] [9, 14, 56, 59, 78, 88] [15, 42, 52, 60, 73, 84] [16, 18, 53, 57, 61, 74] [17, 19, 49, 54, 58, 62] [12, 18, 52, 55, 63, 68] [13, 19, 46, 53, 56, 64] [14, 20, 54, 65, 83, 89] [15, 21, 57, 66, 71, 87] [16, 22, 47, 58, 63, 67] [17, 23, 48, 59, 64, 68] [24, 44, 60, 69, 81, 88] [25, 27, 61, 66, 70, 82] [26, 28, 55, 62, 67, 71] [21, 27, 60, 63, 72, 77] [22, 28, 50, 61, 64, 73] [23, 29, 51, 62, 65, 74] [24, 30, 66, 75, 79, 89] [25, 31, 52, 67, 72, 76] [26, 32, 53, 68, 73, 77] [33, 35, 70, 75, 78, 87] [34, 36, 63, 71, 76, 79] [30, 35, 69, 72, 80, 84] [31, 36, 57, 70, 73, 81] [32, 37, 58, 71, 74, 82] [33, 38, 60, 76, 80, 83] [34, 39, 61, 77, 81, 84] [40, 41, 72, 79, 83, 85] [38, 41, 66, 78, 81, 86] [39, 42, 67, 79, 82, 87] [40, 43, 70, 84, 86, 88] [43, 44, 76, 85, 87, 89]

H_Z (45 checks, sparse supports)

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

Code ID 90-10-7 · download JSON · raw on GitHub